Liverpool striker Suarez has been the subject of a £40 million plus £1 bid by the Gunners, which the Anfield club swiftly rejected.
Dixon, capped 22 times by England during a career in which he twice won the Premier League with Arsene Wenger at Arsenal, admits United's Rooney could begin the campaign with Chelsea, but he is less convinced about Suarez's future.
"I don't think we can nail our colours to the flag with the title race until the transfer window is over," said the 49-year-old, who was speaking while taking part in the inaugural Prudential RideLondon-Surrey 100, cycling 100 miles along the Olympic road race route.
"I still think there is a bit of business to be done, and I think there will be a few big signings just before the end of the deadline.
"I will be interested to see how the big teams start because certainly Manchester City have spent big so far, and the rest are flagging behind as far as spending is concerned. It's probably going to be one of the tightest run races in a long time.
"I think Rooney has a chance of going to Chelsea but I'm not sure about Suarez. I just don't see him going to Arsenal to be honest. I think if he is going to go anywhere it will probably be to Real Madrid."
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