Arsenal tabled a bid for Vardy before he left for the European Championship in France with the rest of England's squad on Monday, according to British media reports.
After the bid from Arsenal, the striker reportedly asked for some time to ponder about his future.
Daily Mail this morning is now reporting that the 29-year-old has verbally agreed the deal of £120,000 a week.
His potential move to Arsenal could spark an exodus of players from Leicester City, Foxes former defender Gerry Taggart has said.
"If Vardy goes then one or two others might want to leave as well," Taggart, who made over 100 appearances for Leicester, told the BBC.
"For the good of the whole team and the club it is important Vardy stays."
"The whole Leicester ethos over the last couple of seasons has been built around people like Vardy," Taggart added.
Vardy scored 24 league goals last season as Leicester won the top-flight crown for the first time in their history, and was voted as the Footballer of the Year by the Football Writers' Association.
The striker along with his teammates on Monday, June 6 arrived France for the Euro 2016.
They open their Group B campaign against Russia on Saturday, June 11 before facing Wales and Slovakia later.