Since 2004, Boston College has retained Fenway Sports Management to help market its athletic programs. But the school has rejected at least one piece of advice from the firm: to host a football game Nov. 19 against the University of Connecticut at Fenway Park.

After last year’s humbling experience at Fenway in a 19-16 loss to Notre Dame, BC appears to be one-and-done with playing football at the home of the Red Sox, at least for now.

The Eagles last year were relegated to serving as the visiting team at Fenway while Notre Dame enjoyed all the perks of hosting the nationally televised installment of its Shamrock Series. BC was allotted only about 5,000 of the 38,686 tickets sold, and the Eagles competed on a field whose end zones were brightly painted with “Notre Dame’’ and “Fighting Irish.’’

This time BC’s role would be reversed. The Eagles would enjoy top billing. They would dress in the Sox clubhouse rather than on campus, as they did last year. Their fans would receive the lion’s share of tickets. Their school would control the end zone displays and much more.

The event could launch an annual tradition of BC hosting games at Fenway.

“Given the positive feedback we received on last November’s Shamrock Series, the Red Sox would love to see a return of BC football to Fenway Park in the near future,’’ Sox president Sam Kennedy said. “We have extended the invitation for BC to return whenever it works for their schedule.’’

The most recent invitation would involve the first game between BC and UConn since 2004, when both schools played in the Big East Conference. The Eagles joined the Atlantic Coast Conference the following year, after UConn had sued BC over the move.

The Sox have proposed that BC shift its scheduled home game against UConn Nov. 19 from Alumni Stadium to Yawkey Way, but BC has rejected the plan. Terms of the offer were not disclosed, but BC athletic director Brad Bates indicated the school has no interest in foregoing anymore home games in Chestnut Hill.

The Sox and Fenway Sports Management are principally owned by Boston Globe publisher John W. Henry.

BC already has agreed to pay Georgia Tech $900,000 to play in the Aer Lingus College Football Classic in Dublin Sept. 3, with BC as the home team. That leaves Boston College with six home games scheduled for Alumni Stadium, the fewest Bates wants to play there.

The Eagles are scheduled to play their second game of the 2016 season as the visiting team at Gillette Stadium against the University of Massachusetts Amherst.

“Boston College often receives requests to play home games at venues ranging from Gillette Stadium to Fenway Park,’’ Bates said. “We have consistently stated that we will consider the possibility only if it is more beneficial for our team, students, and fans, and only during those years in which we still have at least six games in Alumni Stadium. As a result, playing at Fenway Park during the 2016 season was never an option.’’

The allure of playing at Fenway Park includes greater national exposure and perhaps increased ticket sales among patrons who are not typically drawn to BC football but might value the novelty of watching a college game in the iconic ballpark.

Alumni Stadium has a greater seating capacity (44,500) than Fenway (about 39,000 for football), but BC has had trouble filling its home stadium in recent years. The average attendance in Chestnut Hill last year dropped to 30,204, the lowest in 25 years.

However, Fenway Park is not for everyone. BC fans were divided in their reactions to watching the Eagles play there last year. Some complained of poor sight lines. Others missed tailgating on campus. And a few were put off by the high ticket prices.

Boston College, meanwhile, is struggling to meet sales expectations for the Dublin game. The school had projected Eagle supporters would buy at least 10,000 tickets, but the most recent tally released last month by the game’s promoters showed BC fans had bought about 5,000 tickets, surprisingly fewer than the 6,400 purchased by Georgia Tech supporters.

A spokesman for the Dublin game said Thursday he could not provide more precise data. But he said BC appears to be closing the gap on Georgia Tech and estimated that fans of each school would buy at least 7,500 seats before tickets likely sell out when they become available to the general public in Ireland and Europe on April 6.

BC has much to prove in 2016.

The Eagles are coming off a 3-9 season in which they went winless in eight games against ACC opponents.