diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index cd1cb0b64f74a..8a05d0d36cd2f 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -525,6 +525,7 @@ Indexing
 - Bug in :class:`Index` constructor where an unhelpful error message was raised for ``numpy`` scalars (:issue:`33017`)
 - Bug in :meth:`DataFrame.lookup` incorrectly raising an ``AttributeError`` when ``frame.index`` or ``frame.columns`` is not unique; this will now raise a ``ValueError`` with a helpful error message (:issue:`33041`)
 - Bug in :meth:`DataFrame.iloc.__setitem__` creating a new array instead of overwriting ``Categorical`` values in-place (:issue:`32831`)
+- Bug in :class:`Interval` where a :class:`Timedelta` could not be added or subtracted from a :class:`Timestamp` interval (:issue:`32023`)
 - Bug in :meth:`DataFrame.copy` _item_cache not invalidated after copy causes post-copy value updates to not be reflected (:issue:`31784`)
 - Bug in `Series.__getitem__` with an integer key and a :class:`MultiIndex` with leading integer level failing to raise ``KeyError`` if the key is not present in the first level (:issue:`33355`)
 - Bug in :meth:`DataFrame.iloc` when slicing a single column-:class:`DataFrame`` with ``ExtensionDtype`` (e.g. ``df.iloc[:, :1]``) returning an invalid result (:issue:`32957`)
diff --git a/pandas/_libs/interval.pyx b/pandas/_libs/interval.pyx
index a47303ddc93cf..e52474e233510 100644
--- a/pandas/_libs/interval.pyx
+++ b/pandas/_libs/interval.pyx
@@ -1,6 +1,9 @@
 import numbers
 from operator import le, lt
 
+from cpython.datetime cimport PyDelta_Check, PyDateTime_IMPORT
+PyDateTime_IMPORT
+
 from cpython.object cimport (
     Py_EQ,
     Py_GE,
@@ -11,7 +14,6 @@ from cpython.object cimport (
     PyObject_RichCompare,
 )
 
-
 import cython
 from cython import Py_ssize_t
 
@@ -34,7 +36,11 @@ cnp.import_array()
 cimport pandas._libs.util as util
 
 from pandas._libs.hashtable cimport Int64Vector
-from pandas._libs.tslibs.util cimport is_integer_object, is_float_object
+from pandas._libs.tslibs.util cimport (
+    is_integer_object,
+    is_float_object,
+    is_timedelta64_object,
+)
 
 from pandas._libs.tslibs import Timestamp
 from pandas._libs.tslibs.timedeltas import Timedelta
@@ -294,6 +300,7 @@ cdef class Interval(IntervalMixin):
     True
     """
     _typ = "interval"
+    __array_priority__ = 1000
 
     cdef readonly object left
     """
@@ -398,14 +405,29 @@ cdef class Interval(IntervalMixin):
         return f'{start_symbol}{left}, {right}{end_symbol}'
 
     def __add__(self, y):
-        if isinstance(y, numbers.Number):
+        if (
+            isinstance(y, numbers.Number)
+            or PyDelta_Check(y)
+            or is_timedelta64_object(y)
+        ):
             return Interval(self.left + y, self.right + y, closed=self.closed)
-        elif isinstance(y, Interval) and isinstance(self, numbers.Number):
+        elif (
+            isinstance(y, Interval)
+            and (
+                isinstance(self, numbers.Number)
+                or PyDelta_Check(self)
+                or is_timedelta64_object(self)
+            )
+        ):
             return Interval(y.left + self, y.right + self, closed=y.closed)
         return NotImplemented
 
     def __sub__(self, y):
-        if isinstance(y, numbers.Number):
+        if (
+            isinstance(y, numbers.Number)
+            or PyDelta_Check(y)
+            or is_timedelta64_object(y)
+        ):
             return Interval(self.left - y, self.right - y, closed=self.closed)
         return NotImplemented
 
diff --git a/pandas/tests/scalar/interval/test_arithmetic.py b/pandas/tests/scalar/interval/test_arithmetic.py
new file mode 100644
index 0000000000000..5252f1a4d5a24
--- /dev/null
+++ b/pandas/tests/scalar/interval/test_arithmetic.py
@@ -0,0 +1,47 @@
+from datetime import timedelta
+
+import numpy as np
+import pytest
+
+from pandas import Interval, Timedelta, Timestamp
+
+
+@pytest.mark.parametrize("method", ["__add__", "__sub__"])
+@pytest.mark.parametrize(
+    "interval",
+    [
+        Interval(Timestamp("2017-01-01 00:00:00"), Timestamp("2018-01-01 00:00:00")),
+        Interval(Timedelta(days=7), Timedelta(days=14)),
+    ],
+)
+@pytest.mark.parametrize(
+    "delta", [Timedelta(days=7), timedelta(7), np.timedelta64(7, "D")]
+)
+def test_time_interval_add_subtract_timedelta(interval, delta, method):
+    # https://github.com/pandas-dev/pandas/issues/32023
+    result = getattr(interval, method)(delta)
+    left = getattr(interval.left, method)(delta)
+    right = getattr(interval.right, method)(delta)
+    expected = Interval(left, right)
+
+    assert result == expected
+
+
+@pytest.mark.parametrize("interval", [Interval(1, 2), Interval(1.0, 2.0)])
+@pytest.mark.parametrize(
+    "delta", [Timedelta(days=7), timedelta(7), np.timedelta64(7, "D")]
+)
+def test_numeric_interval_add_timedelta_raises(interval, delta):
+    # https://github.com/pandas-dev/pandas/issues/32023
+    msg = "|".join(
+        [
+            "unsupported operand",
+            "cannot use operands",
+            "Only numeric, Timestamp and Timedelta endpoints are allowed",
+        ]
+    )
+    with pytest.raises((TypeError, ValueError), match=msg):
+        interval + delta
+
+    with pytest.raises((TypeError, ValueError), match=msg):
+        delta + interval