diff --git a/nibabel/streamlines/tck.py b/nibabel/streamlines/tck.py
index 823a88b5cf..c9bba94a6e 100644
--- a/nibabel/streamlines/tck.py
+++ b/nibabel/streamlines/tck.py
@@ -8,7 +8,6 @@
 import warnings
 
 import numpy as np
-from numpy.compat.py3k import asbytes, asstr
 
 from nibabel.openers import Opener
 
@@ -44,7 +43,7 @@ class TckFile(TractogramFile):
     .. [#] http://nipy.org/nibabel/coordinate_systems.html#voxel-coordinates-are-in-voxel-space
     """
     # Constants
-    MAGIC_NUMBER = "mrtrix tracks"
+    MAGIC_NUMBER = b"mrtrix tracks"
     SUPPORTS_DATA_PER_POINT = False  # Not yet
     SUPPORTS_DATA_PER_STREAMLINE = False  # Not yet
 
@@ -94,7 +93,7 @@ def is_correct_format(cls, fileobj):
             magic_number = f.read(len(cls.MAGIC_NUMBER))
             f.seek(-len(cls.MAGIC_NUMBER), os.SEEK_CUR)
 
-        return asstr(magic_number) == cls.MAGIC_NUMBER
+        return magic_number == cls.MAGIC_NUMBER
 
     @classmethod
     def create_empty_header(cls):
@@ -230,7 +229,7 @@ def save(self, fileobj):
             header[Field.NB_STREAMLINES] = nb_streamlines
 
             # Add the EOF_DELIMITER.
-            f.write(asbytes(self.EOF_DELIMITER.tobytes()))
+            f.write(self.EOF_DELIMITER.tobytes())
             self._finalize_header(f, header, offset=beginning)
 
     @staticmethod
@@ -251,13 +250,11 @@ def _write_header(fileobj, header):
                    "count", "datatype", "file"]  # Fields being replaced.
 
         lines = []
-        lines.append(asstr(header[Field.MAGIC_NUMBER]))
         lines.append(f"count: {header[Field.NB_STREAMLINES]:010}")
         lines.append("datatype: Float32LE")  # Always Float32LE.
         lines.extend([f"{k}: {v}"
                       for k, v in header.items()
                       if k not in exclude and not k.startswith("_")])
-        lines.append("file: . ")  # Manually add this last field.
         out = "\n".join(lines)
 
         # Check the header is well formatted.
@@ -265,27 +262,24 @@ def _write_header(fileobj, header):
             msg = f"Key-value pairs cannot contain '\\n':\n{out}"
             raise HeaderError(msg)
 
-        if out.count(":") > len(lines) - 1:
+        if out.count(":") > len(lines):
             # : only one per line (except the last one which contains END).
             msg = f"Key-value pairs cannot contain ':':\n{out}"
             raise HeaderError(msg)
 
+        out = header[Field.MAGIC_NUMBER] + b"\n" + out.encode('utf-8')
+
+        # Compute data offset considering the offset string representation
+        # headers + "file" header + END + \n's
+        hdr_offset = len(out) + 8 + 3 + 3
+        offset_repr = f'{hdr_offset}'
+
+        # Adding the offset may increase one char to the offset repr
+        hdr_offset += len(f'{hdr_offset + len(offset_repr)}')
+
         # Write header to file.
-        fileobj.write(asbytes(out))
-
-        hdr_len_no_offset = len(out) + 5
-        # Need to add number of bytes to store offset as decimal string. We
-        # start with estimate without string, then update if the
-        # offset-as-decimal-string got longer after adding length of the
-        # offset string.
-        new_offset = -1
-        old_offset = hdr_len_no_offset
-        while new_offset != old_offset:
-            old_offset = new_offset
-            new_offset = hdr_len_no_offset + len(str(old_offset))
-
-        fileobj.write(asbytes(str(new_offset) + "\n"))
-        fileobj.write(asbytes("END\n"))
+        fileobj.write(out)
+        fileobj.write(f'\nfile: . {hdr_offset}\nEND\n'.encode('utf-8'))
 
     @classmethod
     def _read_header(cls, fileobj):
@@ -320,7 +314,7 @@ def _read_header(cls, fileobj):
             # Read magic number
             magic_number = f.read(len(cls.MAGIC_NUMBER))
 
-            if asstr(magic_number) != cls.MAGIC_NUMBER:
+            if magic_number != cls.MAGIC_NUMBER:
                 raise HeaderError(f"Invalid magic number: {magic_number}")
 
             hdr[Field.MAGIC_NUMBER] = magic_number
@@ -331,7 +325,7 @@ def _read_header(cls, fileobj):
 
             # Read all key-value pairs contained in the header, stop at EOF
             for n_line, line in enumerate(f, 1):
-                line = asstr(line).strip()
+                line = line.decode('utf-8').strip()
 
                 if not line:  # Skip empty lines
                     continue
diff --git a/nibabel/streamlines/tests/test_tck.py b/nibabel/streamlines/tests/test_tck.py
index 1cdda4b44e..75786c87c6 100644
--- a/nibabel/streamlines/tests/test_tck.py
+++ b/nibabel/streamlines/tests/test_tck.py
@@ -216,6 +216,34 @@ def test_write_simple_file(self):
         with pytest.raises(HeaderError):
             tck.save(tck_file)
 
+    def test_write_bigheader_file(self):
+        tractogram = Tractogram(DATA['streamlines'],
+                                affine_to_rasmm=np.eye(4))
+
+        # Offset is represented by 2 characters.
+        tck_file = BytesIO()
+        tck = TckFile(tractogram)
+        tck.header['new_entry'] = ' ' * 20
+        tck.save(tck_file)
+        tck_file.seek(0, os.SEEK_SET)
+
+        new_tck = TckFile.load(tck_file)
+        assert_tractogram_equal(new_tck.tractogram, tractogram)
+        assert new_tck.header['_offset_data'] == 99
+
+        # We made the jump, now offset is represented by 3 characters
+        # and we need to adjust the offset!
+        tck_file = BytesIO()
+        tck = TckFile(tractogram)
+        tck.header['new_entry'] = ' ' * 21
+        tck.save(tck_file)
+        tck_file.seek(0, os.SEEK_SET)
+
+        new_tck = TckFile.load(tck_file)
+        assert_tractogram_equal(new_tck.tractogram, tractogram)
+        assert new_tck.header['_offset_data'] == 101
+
+
     def test_load_write_file(self):
         for fname in [DATA['empty_tck_fname'],
                       DATA['simple_tck_fname']]: