The Day of New London and the Boston Globe, both citing officials involved in the talks, said the game would be played in 2014, and would be considered a home game for the Fighting Irish.
UConn athletic director Warde Manuel did not immediately return an email seeking comment Tuesday morning. Boston Red Sox officials said in March the team was interested in bringing football back to Fenway.
The park has hosted 41 football games in its 100-year history, but none since the New England Patriots beat the Cincinnati Bengals, 33-14, on Dec. 1, 1968.
Notre Dame coach Brian Kelly said the idea is appealing. Yet he is uncertain if Fenway is equipped for football.
"I just saw that. You know me, I love Fenway Park. I just don't know if it's big enough," Kelly said. "As long as they do the due diligence, and I know [Notre Dame athletic director] Jack [Swarbrick] is looking for great venues, and I don't think they played a game there in a long time. If it's on the schedule, we're going to play it.
"Being a Boston guy, baseball has not been very good there, so maybe we'll bring some football."
The last-place Red Sox may be headed toward another offseason of change, but that doesn't mean Kelly has any less desire to play at Fenway.
"I think it would be cool, yeah. I'd like to do it," he said. "But I think more than anything else, I don't know the ramifications of the 2014 schedule as it relates to ACC scheduling and all those kinds of things. I've been so focused on getting to the office today."
But, if UConn and Notre Dame are close to finalizing this deal, it's news to Swarbrick.
"We have work to do on our future football schedules," he said, "given the announcement last week relative to our upcoming Atlantic Coast Conference relationship."
On Nov. 21, 2009, at Notre Dame Stadium, Connecticut outlasted Notre Dame, 33-30, in one of the Huskies' biggest victories as a Division I program.