Alonso has been widely linked with a move to the team he left in 2007 after announcing his departure from Ferrari, and Button hopes to partner him.
The future of the 2009 world champion remains in doubt, though, with McLaren yet to announce their driver line-up for next season.
Button believes he and Alonso could bring the best out of each other and McLaren can have a good season if they develop a competitive car.
"I think every driver likes the challenge of a good team-mate," Button told Crash.net. "That's why I loved the three years I spent with Lewis [Hamilton].
"He was extremely quick, he was better than me in qualifying over the three seasons but I could really challenge him in the race and a lot of times put him in his place in different situations and different conditions.
"So I loved those three years, they were a lot of fun.
"That's what I miss more than anything else right now, it's that fighting for a win. That's what probably is the most difficult thing at this moment in time is that we can't challenge for wins in Formula One.
"So for me if I'm in Formula One next year I've got to be winning, that's what I'm here to do."
Button sits in seventh place in the drivers' standings this season with 94 points, while current team-mate Kevin Magnussen only has 55.