The virtue of Umrah in Ramadan

It is narrated from Sayyiduna Abdullah bin Abbas (may Allah be pleased with them both) that the Messenger of Allah (peace and blessings be upon him) intended to perform Hajj, so a woman said to her husband: "Take me for Hajj with the Messenger of Allah (peace and blessings be upon him)." He said: "I do not have the means to take you for Hajj." She said: "Then take me for Hajj on your water-carrying camel." He said: "My son and I ride that one by turns." She said: "Then take me for Hajj on your such-and-such camel." He said: "That one is dedicated to the path of Allah." She said: "Then sell the dates stored on your shelf." He said: "That is my provision and yours." Then, when the Prophet (peace and blessings be upon him) returned from Makkah, that woman sent her husband to him and said: "Convey my greeting of peace to the Messenger of Allah (peace and blessings be upon him) and ask him: What is equal to performing Hajj with you (peace and blessings be upon him)?" The man came to the Prophet (peace and blessings be upon him) and said: "O Messenger of Allah! My wife sends you greetings of peace and the mercy of Allah. She asked me to take her for Hajj with you, and I said to her: I do not have (the means). She said: Take me for Hajj on your such-and-such camel. I said to her: That one is dedicated to the path of Allah." The Prophet (peace and blessings be upon him) said: "If you had taken her for Hajj, that too would have been in the path of Allah." The narrator said: the Messenger of Allah (peace and blessings be upon him) laughed, marvelling at that woman's eagerness for Hajj. (The man) said: "And she has told me to ask you: What is equal to performing Hajj with you?" He (peace and blessings be upon him) said: "Convey to her my greeting of peace and the mercy of Allah, and tell her that performing Umrah in Ramadan is equal to performing Hajj with me."
This hadith is sahih (authentic) on the condition of the two Shaykhs (Bukhari and Muslim), but they did not record it.

It is narrated from the Mother of the Believers, Sayyidah Aishah (may Allah be pleased with her), that in the year of the Farewell Hajj the Messenger of Allah (peace and blessings be upon him) commanded the people, saying: "Whoever would like to perform Umrah before Hajj and come out (of ihram), let him do so."
This hadith has a sahih (authentic) chain of narration, but the two Shaykhs did not record it.