Mussar- Rachamim

Compassion depends upon the internal connectedness that arises from a sense of shared identity, as the you and the me are mingled in a oneness that transcends our perceptions of separate identities.
Morinis, Alan. Everyday Holiness: The Jewish Spiritual Path of Mussar (p. 79). Shambhala. Kindle Edition.

פסחים קט״ז ב:ה׳

בכל דור ודור חייב אדם לראות את עצמו כאילו הוא יצא ממצרים שנאמר (שמות יג, ח) והגדת לבנך ביום ההוא לאמר בעבור זה עשה ה' לי בצאתי ממצרים

Pesachim 116b:3

In each and every generation one is obligated to see themselves as if they went out from Egypt, as it says (Exodus 13:8) And you shall tell you child on that day, saying: Because of this, God did for me when I went out from Egypt.

Compassion may indeed involve feeling. It also may breed action. But these qualities do not take the definition far enough. The soul-trait of compassion may be more accurately defined as the inner experience of touching another being so closely that you no longer perceive the other one as separate from you.
Morinis, Alan. Everyday Holiness: The Jewish Spiritual Path of Mussar (p. 80). Shambhala. Kindle Edition.

Rachamim—compassion—does not come into being just by feeling empathy. The depth and richness of the emotional connection must be translated into action that expresses concretely how truly moved you are to take care of the other.
Morinis, Alan. Everyday Holiness: The Jewish Spiritual Path of Mussar (p. 81). Shambhala. Kindle Edition.

Though it is not its defining characteristic, compassion does have an important emotional component. My oneness with you means that whatever you are feeling is also stirred within me as my own emotional experience. Your sadness is my sadness. Your pain is my pain. Your confusion is my confusion. Your joy is my joy. Compassion may not be an emotion, but it cannot exist without full emotional contact with the other.
Morinis, Alan. Everyday Holiness: The Jewish Spiritual Path of Mussar (p. 81). Shambhala. Kindle Edition.

בראשית רבה י״ב:ט״ו

(טו) ה' אֱלֹהִים, לְמֶלֶךְ שֶׁהָיוּ לוֹ כּוֹסוֹת רֵיקִים, אָמַר הַמֶּלֶךְ אִם אֲנִי נוֹתֵן לְתוֹכָן חַמִּין, הֵם מִתְבַּקְּעִין. צוֹנֵן, הֵם מַקְרִיסִין, וּמֶה עָשָׂה הַמֶּלֶךְ עֵרַב חֲמִין בְּצוֹנֵן וְנָתַן בָּהֶם וְעָמָדוּ. כָּךְ אָמַר הַקָּדוֹשׁ בָּרוּךְ הוּא אִם בּוֹרֵא אֲנִי אֶת הָעוֹלָם בְּמִדַּת הָרַחֲמִים, הֲוֵי חֶטְיָיה סַגִּיאִין. בְּמִדַּת הַדִּין, הָאֵיךְ הָעוֹלָם יָכוֹל לַעֲמֹד. אֶלָּא הֲרֵי אֲנִי בּוֹרֵא אוֹתוֹ בְּמִדַּת הַדִּין וּבְמִדַּת הָרַחֲמִים, וְהַלְּוַאי יַעֲמֹד.

Bereshit Rabbah 12:15

(15) To what might God be compared? A king who had empty glasses. The king said "if I put hot water in them, then they will expand and break, and if I put cold water in them, they will contract and shatter. What did the king do? He mixed hot water with the cold water and put them in the glasses. So too the Holy One of Blessing said: if I create the world with the attribute of compassion alone, no one would be concerned with the consequences of their actions. With the attribute of judgment alone, how could the world stand? Rather, behold I create it with both the attribute of judgment and the attribute of compassion, and hopefully it will stand.

ברכות ז׳ א:א׳-ג׳

אָמַר רַבִּי יוֹחָנָן מִשּׁוּם רַבִּי יוֹסֵי: מִנַּיִן שֶׁהַקָּדוֹשׁ בָּרוּךְ הוּא מִתְפַּלֵּל? שֶׁנֶּאֱמַר: ״וַהֲבִיאוֹתִים אֶל הַר קָדְשִׁי וְשִׂמַּחְתִּים בְּבֵית תְּפִלָּתִי״, ״תְּפִלָּתָם״ לֹא נֶאֱמַר, אֶלָּא ״תְּפִלָּתִי״, מִכָּאן שֶׁהַקָּדוֹשׁ בָּרוּךְ הוּא מִתְפַּלֵּל.

מַאי מְצַלֵּי? אָמַר רַב זוּטְרָא בַּר טוֹבִיָּה, אָמַר רַב: ״יְהִי רָצוֹן מִלְּפָנַי... וְאֶתְנַהֵג עִם בָּנַי בְּמִדַּת רַחֲמִים.

Berakhot 7a:1-3

Rabbi Yoḥanan said in the name of Rabbi Yosei: From where is it derived that the Holy One, Blessed be He, prays? They give a prooftext from scripture: “I will bring them to My holy mountain, and make them joyful in the house of My prayer” (Isaiah 56:7). The verse does not say the house of their prayer, but rather, “the house of My prayer”; from here we see that the Holy One, Blessed be He, prays.

The Gemara asks: What does God pray? Rav Zutra bar Tovia said that Rav said:
God says: May it be My will that... I conduct myself toward My children, Israel, with the attribute of compassion (rachamim).

שבת קנ״א ב:י״ד

רבי גמליאל ברבי אומר ונתן לך רחמים ורחמך והרבך כל המרחם על הבריות מרחמין עליו מן השמים וכל שאינו מרחם על הבריות אין מרחמין עליו מן השמים

Shabbat 151b:14

Rabbi Gamliel, son of Rabbi Yehuda HaNasi, says: The verse that states: “And He will show you mercy and have compassion on you and multiply you” (Deuteronomy 13:18) teaches us that anyone who has compassion for God’s creatures will receive compassion from Heaven, and anyone who does not have compassion for God’s creatures will not receive compassion from Heaven.