A game of chicken revolves around Sonny Gray as teams wonder if the A’s will take the best offer available by 4 p.m. Monday or hold the righty and market him again in the offseason if they do not get their desired package.

Oakland officials have let it be known they will not undersell the righty and always can revisit trading Gray when the inevitable free-agent sticker-price shock for top starters occurs in November and December.

Nevertheless, interested teams recognize Oakland would be taking a risk. The biggest concern about Gray is his ability to stay healthy, which he is right now and pitching well. He allowed four runs, but none earned, and struck out six Tuesday night in a loss to Toronto.

That is Gray’s sixth straight start of two or fewer earned runs allowed. If they are going to move him, the A’s would almost certainly prefer not to incur the peril of letting him make his next scheduled start, Sunday, the day before the deadline.

That is why the perception is growing in the industry of Oakland pushing for final and best offers from teams, with the belief the Yankees are sitting on the best bid and the A’s wanting to see if another club will top it or the Yankees will push even more into their package.

As if this one start did not carry enough risk, if the A’s do not trade Gray by the deadline, they would need him to make about a dozen more starts before the offseason without breaking down.

Remember that the A’s moved Sean Doolittle to Washington 15 days before the deadline rather than see whether the clock increased offers, in part, because they were worried about the lefty’s fragility.

In addition, whatever the prices, free-agent starters such as Jake Arrieta, Yu Darvish, Alex Cobb, Lance Lynn, etc., will only cost money, and Japanese star Shohei Otani also could be part of the market — all of which could complicate moving Gray.

So Oakland could let the main contenders for Gray — the Yankees, Brewers, Braves, Astros and, perhaps, Dodgers — make offers and see if the ticking clock motivates better packages closer to the deadline, but know they are going to take the best return before the non-waiver period ends.

The four most interested teams all have strong farm systems.

Milwaukee, for example, was willing to include their 2016 first-round pick, outfielder Corey Ray, and pitcher Luis Ortiz, a key return from Texas last year for Jonathan Lucroy, in offers for Jose Quintana, and might listen on one or both for Gray, sources told The Post.

Ray would fit one element the A’s are looking for — a center fielder of the future. Finding one is not a deal breaker, sources told The Post, but it is a preference.

One of the A’s most trusted evaluators — director of player personnel Billy Owens — has been canvassing the Yankees organization and has seen both Estevan Florial, at Single-A, and Jorge Mateo, at Double-A, play center field. Florial’s star has been on the rise this year (he played in the Futures Game) and he is considered a player who can stay in and excel at center field.

Mateo has revived some of his once-lofty prospect status by performing well after his elevation to Double-A, flashing both his top-of-the-scale foot speed and high-end bat speed. Mateo is a natural shortstop, but scouts who have seen him say center field may be his best position due not just to his legs, but better instincts for the position than anticipated.

Owens also recently saw well-regarded Domingo Acevedo start. Six-foot-eight righty starter Freicer Perez has been getting a lot of love from scouts in low-A, as has Albert Abreu, the main piece the Yanks received from Houston for Brian McCann.

The Yankees believe they upgraded their chances to make the playoffs — and be a force should they get in — by improving their bullpen with Tommy Kahnle and David Robertson. They would love to add a starter to hedge against the fragility of CC Sabathia and Masahiro Tanaka, and the youth/endurance of Jordan Montgomery and Luis Severino.

They feel they made a strong offer for their No. 1 target, Quintana, since he checked the boxes of being in his prime, left-handed and in possession of a team-friendly contract for years to come.

The A’s are said to want a deal structured similarly to what the White Sox received for Quintana — two high-end prospects and two pieces perhaps further away, but with upside.

The Yankees do not value Gray to the same level, in part because of his durability questions. But he is controllable through 2019, which is appealing to the Yankees, and he is their main aim now even over the more talented Yu Darvish.

Darvish can be a free agent after the season, Texas is requesting a huge return to part with him and there are many questions within the game about Darvish acclimating somewhere else quickly, in part because he has such affection for being a Ranger.

The Yankees do seem determined to add a starter just to create more options for Joe Girardi and greater depth to hedge against injury. So if it were not Gray, they would have to look to a different level (but with a lower asking price). Starters such as Toronto’s Marco Estrada, St. Louis’ Lance Lynn and Colorado’s Tyler Chatwood are all in their walk years.