The Spaniard will this week rejoin his McLaren-Honda F1 team after a fortnight in Indianapolis culminating in Sunday's Indy 500.
Alonso finished 24th after retiring from seventh with a Honda engine failure, the same type of problem that convinced him to skip the Monaco Grand Prix.
However, he will be back in the seat for the Canadian Grand Prix in two week's time although it could be his last trip to Montreal as a McLaren driver.
His contract with the team expires at the end of the current season and the 35-year-old is unlikely to re-sign given the car's lack of pace so far this year.
Fernando Alonso at the Indy 500 Wed, May 3, 2017
Fernando Alonso has tested his McLaren Honda Andretti Indycar ahead of the Indianapolis 500
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Fernando Alonso shows off his new McLaren Honda Andretti Indycar
But the two-time world champion is still regarded as one of the grid's most talented drivers and is understood to have a number of teams ready to court him this summer.
Alonso has ruled out one team though – Red Bull's success in promoting young drivers such as Sebastian Vettel, Daniel Ricciardo and Max Verstappen would make signing Alonso an unscripted move.
But having raced for McLaren, Ferrari and Renault previously, the enigmatic driver is going into the summer with an open mind.
"We'll see. I would be lying if I told you now that I have a concrete plan," Alonso told Spanish outlet Planeta Calleja.
"I could go to another team – I don't know whether it is Mercedes or another.
"If Renault starts to dominate, I don't know.
"If in June or July a team calls, I think we would have a chance, but it's all to be talked about.
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"I think perhaps Red Bull is the only one that has doors a little closed because it already has young drivers with long contracts."
Fernando Alonso reckons Red Bull's doors are already closed to him for next year
Fernando Alonso is unlikely to re-sign for McLaren
Alonso also ruled out retiring with his continued passion for racing evident during his recent Indy 500 stint.
But he did suggest that if new owners of F1 Liberty Media extend the season much further, he'd be tempted to hang up his helmet.
"I read that they want to make a championship of 25 races," Alonso added.
"When I started in F1 there were 16, now there are 20.
"If they tell me there are 25 I will retire!"