Undoubtedly, Patriots coach Bill Belichick would have made every effort to get more of the former than the latter out of Peppers, who was the prize of the free agent market last offseason.

There had long been rumors that the Patriots wanted Peppers, and last season there were rumors he might be traded to New England. But when midnight struck March 5, Peppers and the Chicago Bears quickly finalized a six-year contract, with $42 million guaranteed.

But Peppers acknowledged that he was considering the Patriots.

“I thought about a number of teams, including New England,’’ Peppers said. “During free agency, you’ve got to look at all your options and you want to play for one of the better franchises. I definitely had a little interest in playing for those guys.’’

When asked what tipped things in the Bears’ favor, the answer Peppers gave was interesting.

“It was more scheme, coaches, players,’’ he said. “I’ve got a lot of respect for this [Chicago] team — the coaches and the players — so I couldn’t pass on the opportunity to play with these guys.’’

Belichick theorized that the 6-foot-6-inch Peppers — a former power forward on the North Carolina basketball team — could play almost any position on the field save cornerback.

“You have to account for him on every play,’’ said Belichick. “They’ve got a lot of good guys on that front, they’re all good, and the linebackers are good. So it’s definitely not a one-man band. But you have to know where Peppers is on every play.

“A lot of times, he’s being disruptive, and that ends up with other guys making plays. Or the other guys on the front are being disruptive and he runs them down.

“But he’s a tremendous athlete. He’s got a good motor, very hard to block, great balance, quickness, speed, his length, tipping balls, getting in the passing lane, and all that.’’

There were reports that Peppers did not want to play in a 3-4 defense, which is the Patriots’ base alignment. But both sides seem happy with the way things have turned out.

Bears coach Lovie Smith called Peppers, who has seven sacks, an impact player.

“I just couldn’t be more pleased with what he’s done for our team,’’ Smith said.

“You play for years in high school, you play for years in college, and you get to do this job that every kid dreams of,’’ Graham said. “I think no matter where you get to play, what the situation is for the team, it’s a dream job.

“But it’s really nice to be a part of this team and then the city and the aura and the atmosphere, and especially being part of a team that’s motivated and likes to win and does the right things to win. So that makes it that much more of a dream job to be able to play here.’’

Graham, who turned 33 yesterday, played with the Bills, Panthers, Bengals (for seven seasons), and one game this year with the Giants.

He was signed Nov. 10 as an injury replacement for Stephen Gostkowski and has made all five of his field goal attempts — the longest was 41 yards Monday against the Jets — and 19 of 20 extra points.

“Vertical threat, good speed, good receiving tight end,’’ said Patriots safety James Sanders. “He’s a big guy, runs well. We’re going to have our hands full with him.

“We’ve got some of the best tight ends, I feel, in the league and we get a good look each and every day, and they prepare us as much as possible leading into the game.”

Greg A. Bedard of the Globe staff contributed to this report. Shalise Manza Young can be reached at syoung@globe.com. Follow her on Twitter @shalisemyoung.