The Gunners have triggered a release clause in the England international's contract, and are believed to have offered him a £120,000-a-week deal.
Premier League champions Leicester are thought to have made Vardy, 29, a counter offer as they try to keep him.
Arsenal, whose bid is reported to be about £20m, were expecting a response from the player on Monday.
He is now in France with England and is said to be fully focused on Euro 2016.
Manager Roy Hodgson has made it clear he would not welcome club matters imposing on England's campaign, which starts against Russia in Marseille on Saturday.
Vardy's camp have not told Arsenal he will delay his decision until after the Euros, so the north London club have not given up hope of him agreeing to join beforehand.
They are not pulling out of the deal and their offer stands, but there is a good chance any move may not now happen until later in the summer.
Vardy, whose 24 goals helped Claudio Ranieri's side become English champions, joined Leicester from Fleetwood for £1m in 2012.
The League One side are set to make in the region of £4m if Vardy does move on, after negotiating a sell-on clause with Leicester.
Fleetwood chairman Andy Pilley said National League North side FC Halifax Town, another of Vardy's former clubs, will "benefit significantly" if the transfer goes through.
"I would be lying if I said I wasn't watching events closely," Pilley told BBC Radio 5 live. "Financially it would be a boost.  I think at some point Jamie will move on. It is Roy of the Rovers stuff."
Vardy, the Football Writers' Association Footballer of the Year for 2015-16, has scored three times in eight games for England.
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Leicester City striker Jamie Vardy's representatives have told Arsenal the striker wants to focus on Euro 2016.