Masaomi meets Anri through Mikado and the three of them develop a fast friendship together. The trio are often seen hanging out either to see the sights or to participate in whatever crazy Idea Masaomi is able to think up at the time (as shown in episode 12.5). Specifically with Anri and Masaomi, despite the relatively short time they have known each other, they have developed a very close bond. When Anri was attacked by Haruna, Masaomi was willing to return to the Yellow Scarves, a part of his life he was desperately trying to leave behind, just to get revenge on the slasher. Likewise, Anri sees Masaomi, as well as Mikado, as the people who have finally managed to draw her out of her emotional shell that she encased herself in since her parents' deaths. She values her friendship with both of them deeply and (in the anime) and is willing to go to extreme measures to prevent them from being harmed.
Anri aided in saving Masaomi from Horada and his rebellion through her children.
It is also worth noting that Masaomi has been known to be rather flirtatious with Anri at various points in the series. However, considering that Masaomi is flirtatious with almost every woman he meets, as well as his relationship with Saki, it's best to assume these advances aren't serious.
Both also grew concerned about Mikado's behavior and try to talk to him.
Some strain is placed on their relationship towards the end of the anime after Anri discovers Masaomi was the leader of the Yellow Scarves. Likewise, Masaomi becomes infuriated when he figures out that Anri is affiliated with the Dollars as well as the fact that she was seen spying on him and his group. This culminates with him confronting Anri about her intentions and he even accuses her of manipulating him and Mikado for her own ends which prompts her to slap him. Within the anime, the conflicts are brought to an end with the help of Celty and Kadota and the trio are able to resolve their differences peacefully, although Masaomi still decides to leave the city.
In the light novels, however, it is revealed the Masaomi has been rather afraid of returning to the city, as he felt that the others were still mad at him over the incident with the Yellow Scarves, or that too much time had passed and, if he were to return, things wouldn't be the same between them. Some encouraging words from Saki dispel any doubts Masaomi had and he resolves to return to Ikebukuro.