Is there a hadith about the daughter of Imam Ali (As) who was asked in marriage by the second caliph Umar?  Did a marriage take place at all?
Answer
Bismillah al-Rahmaan al-Raheem
The short answer to this question is yes there are hadiths that attempt to suggest that Umar bin al-Khattaab asked for the daughter of Imam Ali (AS). The hadiths that are reported in such sunny hadith books as the book of al-Estee'aab by Ibn Abd-el-Birr, the book of Taareekh Baghdaad by al-Khateeb al-Baghdaadi, and the book of al-Esaabah by Ibn Hajar are quite clearly false and un-authentic; not only on the basis of the narrators of the hadiths, but also on the basis of the content of the hadiths.
In one hadith "Umar asks Imam Ali (AS) for his daughter in marriage, and Imam Ali (AS) asks him to leave and then Imam Ali asks his daughter to go to Umar, and when she arrives Umar asks her to marry him, and places his hand on her leg and uncovers her leg!!!!!!  She immediately says to him you do this to me!  If you were not the ruler I would have broken your nose. And she leaves the room. She tells Imam Ali of what happened and added 'You sent me to a vile old man.'  But Ali nevertheless marries her to him!"  al-Estee'aab  4/1954.
In another hadith "Umar asks Imam Ali (AS) for his daughter in marriage, and Imam Ali (AS) asks him to leave and then Imam Ali asks his daughter to make herself beautiful to go to Umar!!!  And when Umar saw her he placed his hand on her leg and asked her to marry him . . . when she came back to her father Imam Ali asked her what did Umar tell you, she replied Umar kissed me and placed his hand on my leg and asked me to marry him.  Then Ali married her to him!!!!!"  Taareekh Baghdaad 6/182.
Ibn Hajar reports in his book al-Esaabah that "Umar asks Imam Ali (AS) for his daughter Umm Kolthum (AS) in marriage, and Imam Ali (AS) asks him to leave and then Imam Ali asks his daughter to go to Umar, and when she arrives Umar placed his hand on her leg and uncovers her leg!!!!!!  The daughter of Imam Ali (AS) says to him if you were not the ruler I would have gouged your eyes . . . ." al-Esaabah 4/321.
The behaviour of Umar according to such hadiths has been so fouls and abhorrent that many sunny scholars have been put to shame.  Because such behaviour is not only disgraceful, but it is also totally haram according to Islam.  One of the top sunny scholars, Ibn Jawzey, states in his book "Tathkirat Khawas al-Ummah" p 321: "And by Allah this is vile and despicable, even if she was a slave Umar may not have done this to her, and furthermore in accordance with the unanimous consensus of the scholars of the Muslims, the touching of a non-Mahram is not allowed, then how this hadith can be attributed to Umar?"
As you can see these hadiths are totally false and any respectable and honourable Muslim would not send his daughter to the house of another man or agree to such abhorrent behaviour let alone agreeing to marrying his daughter.  These hadiths have been placed by the enemies of Imam Ali (AS) and the enemies of the Ahl-ul-Bayt (AS) in order to demean Imam Ali and his household (AS) and give some status to Umar.  Such a marriage certainly did not take place at all.
Imam Ali, the son of Abi Talib (peace be upon them), had two daughters from Lady Fatimah al-Zahra' peace be upon her, the beloved daughter of the holy Prophet Muhammad may Allah bless him and his pure family. They were Zaynab and Umm_Kolthoum peace be upon them.
Lady Zaynab married her cousin Abdullah the son of Ja'far ibn Abi_Talib - also known as Ja'far al-Tayyar. Ja'far al-Tayyar, who on the orders of the prophet headed the Muslims who migrated to Abyssinia , is the brother of Imam Ali peace be upon them.
Lady Umm_Kolthoum married her cousin Muhammad/Sa'id the son of Aqeel ibn Abi_Talib. Aqeel ibn Abi Talib is the elder brother of Imam Ali ibn Abi Talib.