Nasri has been linked with a move to Premier League champions Manchester United but Dixon has warned the Gunners it would be a grave mistake to say goodbye to two of their biggest stars.
Dixon, who played for Arsenal from 1988 until 2002, told The Sun: "Every year it's a case of is Cesc going or is he staying?
"It will be no different this summer and it's inevitable that he will be linked with a move to Spain. At some point, it's also inevitable that he will actually leave.
"With Nasri, he had a great start to the season and I think he benefitted when Cesc wasn't in the team. He seemed to come into his own game a lot more in a central role instead of being out on the wing.
"I think it's imperative the manager keeps the best players, but do you want to keep someone if they are unhappy? In my book, it's always best to let unhappy players go and I think Cesc will come into that category sooner rather than later.
"I said at the start of last season that if Arsenal didn't start well, you could quite easily see his head dropping. I think he got the captaincy as a result of that to try and keep him at the club and inspire him to push the team on to win trophies.
"But that didn't happen and I don't particularly think he's a good captain anyway. I think it's imperative Nasri is made to feel welcome and shown that new signings will come in to strengthen the team.
"Will Cesc go? I think he'll definitely go at some point but to lose both of them will be a big, big blow."