New York had hoped Eovaldi might be able to rejoin the rotation if New York reaches the playoffs, but Girardi acknowledged that prospect is less likely.
Girardi said he hopes Eovaldi will be available to pitch in relief about one week after he is able to resume throwing.
Eovaldi, who is 14-3 with a 4.20 ERA in 27 starts, hasn't pitched since Sept. 5 against Tampa Bay. Two days later, an MRI confirmed inflammation in his elbow.
Eovaldi allowed eight runs and 13 hits over 10 1-3 innings in two starts before his injury was diagnosed.
Right-hander Masahiro Tanaka, who has a strained right hamstring, did conditioning drills and played catch at Yankee Stadium on Monday. New York hopes he can return next week after missing Wednesday's scheduled start against the AL East-leading Blue Jays. Ivan Nova will start in Tanaka's place against Toronto.