"Therefore the children of Israel do not eat the sinew of the thigh-vein" (Genesis 32:33). We learned: A person may send to a gentile a thigh that has the sinew of the thigh-vein within it. Rabbi Abbahu said: Wherever it says "they shall not eat," both a prohibition of eating and a prohibition of benefit are implied, until Scripture spells it out for you as it spelled out for you concerning carrion. Rabbi Abbahu holds that when carrion was permitted [for benefit], its fat and its sinew were permitted. According to Rabbi Yehudah, who holds that there is a flavor-imparting quality in the sinews, [the sinew is forbidden to eat]; and according to Rabbi Shimon, who holds there is not, he is of the opinion that the sinew of the thigh-vein is forbidden in benefit.
May One Give the Forbidden Thigh-Sinew to a Gentile
Yalkut Shimoni on Torah 133:7
עַל כֵּן לֹא יֹאכְלוּ בְנֵי יִשְׂרָאֵל אֶת גִּיד הַנָּשֶׁה. תְּנַן, שׁוֹלֵחַ אָדָם לְנָכְרִי יֶרֶךְ שֶׁגִּיד הַנָּשֶׁה בְּתוֹכָהּ רַבִּי אַבָּהוּ אָמַר כָּל מָקוֹם שֶׁנֶּאֱמַר לֹא יֹאכְלוּ אֶחָד אִסּוּר אֲכִילָה וְאֶחָד אִסּוּר הֲנָאָה בְּמַשְׁמַע, עַד שֶׁיִפְרֹט לְךָ הַכָּתוּב כְּדֶרֶךְ שֶׁפָּרַט לְךָ בִּנְבֵלָה קָסָּבַר רַבִּי אַבָּהוּ כְּשֶׁהֻתְּרָה נְבֵלָה חֶלְבָּהּ וְגִידָהּ הֻתְּרוּ לְרַבִּי יְהוּדָה דְּאִית לֵיהּ יֵשׁ בְּגִידִים בְּנוֹתֵן טַעַם וּלְרַבִּי שִׁמְעוֹן דְּלֵית לֵיהּ סְבִירָא לֵיהּ דְּגִיד הַנָּשֶׁה אָסוּר בַּהֲנָאָה.