diff --git a/scipy/odr/odrpack.py b/scipy/odr/odrpack.py
index 878327d0f90b..45fb08c91dcf 100644
--- a/scipy/odr/odrpack.py
+++ b/scipy/odr/odrpack.py
@@ -35,6 +35,7 @@
 robert.kern@gmail.com
 
 """
+import os
 
 import numpy
 from warnings import warn
@@ -656,11 +657,13 @@ class ODR(object):
         ODRPACK User's Guide if you absolutely must set the value here. Use the
         method set_iprint post-initialization for a more readable interface.
     errfile : str, optional
-        string with the filename to print ODRPACK errors to. *Do Not Open
-        This File Yourself!*
+        string with the filename to print ODRPACK errors to. If the file already
+        exists, an error will be thrown. The `overwrite` argument can be used to
+        prevent this. *Do Not Open This File Yourself!*
     rptfile : str, optional
-        string with the filename to print ODRPACK summaries to. *Do Not
-        Open This File Yourself!*
+        string with the filename to print ODRPACK summaries to. If the file
+        already exists, an error will be thrown. The `overwrite` argument can be
+        used to prevent this. *Do Not Open This File Yourself!*
     ndigit : int, optional
         integer specifying the number of reliable digits in the computation
         of the function.
@@ -709,6 +712,9 @@ class ODR(object):
     iwork : ndarray, optional
         array to hold the integer-valued working data for ODRPACK. When
         restarting, takes the value of self.output.iwork.
+    overwrite : bool, optional
+        If it is True, output files defined by `errfile` and `rptfile` are
+        overwritten. The default is False.
 
     Attributes
     ----------
@@ -725,7 +731,8 @@ class ODR(object):
     def __init__(self, data, model, beta0=None, delta0=None, ifixb=None,
         ifixx=None, job=None, iprint=None, errfile=None, rptfile=None,
         ndigit=None, taufac=None, sstol=None, partol=None, maxit=None,
-        stpb=None, stpd=None, sclb=None, scld=None, work=None, iwork=None):
+        stpb=None, stpd=None, sclb=None, scld=None, work=None, iwork=None,
+        overwrite=False):
 
         self.data = data
         self.model = model
@@ -743,6 +750,13 @@ def __init__(self, data, model, beta0=None, delta0=None, ifixb=None,
         if ifixx is None and data.fix is not None:
             ifixx = data.fix
 
+        if overwrite:
+            # remove output files for overwriting.
+            if rptfile is not None and os.path.exists(rptfile):
+                os.remove(rptfile)
+            if errfile is not None and os.path.exists(errfile):
+                os.remove(errfile)
+
         self.delta0 = _conv(delta0)
         # These really are 32-bit integers in FORTRAN (gfortran), even on 64-bit
         # platforms.
diff --git a/scipy/odr/tests/test_odr.py b/scipy/odr/tests/test_odr.py
index f58f0190e7b9..75a4b942701a 100644
--- a/scipy/odr/tests/test_odr.py
+++ b/scipy/odr/tests/test_odr.py
@@ -1,4 +1,7 @@
 # SciPy imports.
+import tempfile
+import shutil
+import os
 import numpy as np
 from numpy import pi
 from numpy.testing import (assert_array_almost_equal,
@@ -495,3 +498,25 @@ def func(par, x):
         sd_ind = out.work_ind['sd']
         assert_array_almost_equal(out.sd_beta,
                                   out.work[sd_ind:sd_ind + len(out.sd_beta)])
+
+    def test_output_file_overwrite(self):
+        """
+        Verify fix for gh-1892
+        """
+        def func(b, x):
+            return b[0] + b[1] * x
+
+        p = Model(func)
+        data = Data(np.arange(10), 12 * np.arange(10))
+        tmp_dir = tempfile.mkdtemp()
+        error_file_path = os.path.join(tmp_dir, "error.dat")
+        report_file_path = os.path.join(tmp_dir, "report.dat")
+        try:
+            ODR(data, p, beta0=[0.1, 13], errfile=error_file_path,
+                rptfile=report_file_path).run()
+            ODR(data, p, beta0=[0.1, 13], errfile=error_file_path,
+                rptfile=report_file_path, overwrite=True).run()
+        finally:
+            # remove output files for clean up
+            shutil.rmtree(tmp_dir)
+