On Friday [12 January], reports suggested Jose Mourinho has identified the 31-year-old as one of three attacking targets during the January transfer window.
But according to the Leicester Mercury, United have no plans to firm up their interest with a bid.
Initial suggestions were that the Foxes would demand £35m for the ex-Fleetwood Town hitman, who turned down a move to Arsenal in the summer of 2016.
But latest reports claim that it could take an offer in excess of the £75m that United paid to sign Romelu Lukaku from Everton in the summer, to acquire Vardy's services.
There were already concerns that Vardy's age could see United's executive vice-chairman Ed Woodward block the transfer given he would hold no resale value.
And that stance is likely to have been strengthened by Leicester's desire not to sell unless the 20-time English champions submit an "extraordinary" offer for Vardy.
The 2015-16 champions are prepared to back manager Claude Puel in keeping their best players, though it remains to be seen whether they can keep ahold of winger Riyad Mahrez.
Though Vardy endured a poor campaign as Leicester produced a limp defence of their league title last term, he has returned to form with 10 goals in 23 appearances.
He scored in the 2-2 draw with United in December to become the first Leicester player to reach 50 goals in the Premier League, having previously scored in a record 11 consecutive games in their title-winning campaign,
Vardy is also destined to be included in Gareth Southgate's England squad for this summer's World Cup in Russia.