In the movie, Crouching Tiger, Hidden Dragon, we could tell that Li Mu Bai had a strong relationship with his master. We could see that he looked to his master as a father figure and maybe even family. It is clear that to be called someones master is a very high honor.
Li Mu Bai had a strong desire to teach Jen because he saw and understood that she had the potential to be a great swordsman. He saw that she had great skill with the Green Destiny but her skills had room for improvement. Li Mu Bai also said to Jen that he really wanted to have an apprentice or pupil to teach. I believe that Li Mu Bai saw that Jen could not be corrupted by Jade Fox and that her skills can continue to excel. We can see that Jen has surpassed her current master Jade Fox in ability and can still improve even more. In a personal note, I think having an apprentice is a way of having immortality. You teach your pupils your ways and motives and people recognize and know that the apprentices style and technique is yours. If this were true, an added bonus to this would be that Li Mu Bai would be ending Jade Fox's potential immortality if he could teach Jen. We could tell that Jen kind of wanted Li Mu Bai to teach her too. After he asks Jen if he could teach her about the ways of the Wudang style, we could see that she was thinking about it a little bit before she ran away. At the end of the movie, Jade Fox explains that the poison dart was meant for Jen, her own protegee, not for Li Mu Bai. If Li Mu Bai didn't get hit, he would have had the perfect opportunity to teach Jen.