The 7 May match against Everton, which could see the Foxes become champions, sold out in 90 minutes on Monday.
But within hours, individual tickets were being advertised for more than £3,000 each and a pair for much more.
The club said it would “take action against any ticket touting” or attempts to “resell tickets above face value”.
Leicester are seven points clear of nearest rivals Tottenham with five games left. They have already secured Champions League football and need three more wins to guarantee the Premier League title.
They face West Ham and Swansea at their King Power Stadium then Manchester United at Old Trafford, before the game against Everton next month, followed by the final match of the season away to last year’s champions Chelsea on 15 May.
One fan who contacted BBC Sport said he had missed out on a ticket, despite having a membership package and having been to all but three home games this season.
“This season is something we never expected and will probably never happen again in our lifetimes,” said the supporter, who wished to remain anonymous.
“Now I won’t be there to potentially celebrate us winning the Premier League. It is a bit of a ‘here’s what you could have won’ moment.
“The club have been brilliant with fans these past couple of seasons. But I think there will be a few questions for them to answer as to how this has happened. I can’t see how ordinary fans could afford to pay £15,000 for two tickets – it’s the price of a second-hand car.”
Leicester posted on their own website that the Everton game had seen “unprecedented demand” and sold out in “record time”, with the “vast majority” of higher-tier members able to get a ticket.
Membership does not guarantee a ticket, the club told the Leicester Mercury.
“The unauthorised sale of football tickets is a criminal offence under UK law,” added a club spokesman.
A number of fans responded to the club’s Facebook post with their complaints:
David Lea: “Very bad when they are on ticket sites at anything from £495 to £5,000. Who’s giving the touts these tickets? What’s the point of being a gold/silver member? I’d like to hear the club’s views on this.”
Ben Dunkley: “Absolute joke, what a waste of money spending £75 on a gold membership.”
Peter Schoneveld: “I made 85 calls since 9am as I have to book by phone because I have a family membership and was unable to get through. The system is rubbish. I have only been able to get tickets to one match all season!! Why am I paying over £100 for a membership (plus however much on phone calls sitting through an automated message to then be cut off) to get priority on tickets and still unable to get any! I am gutted… and angry.”
Emilia Woch: “The shameful fact of the matter is half these people who got tickets will go on to sell them off for thousands of pounds, whilst us TRUE fans (who find it hard enough to afford £50 a ticket, let alone thousands of pounds) miss out. If the club truly cares about its fans, it would void every resold ticket and sell them back to the fans who don’t want to make money off Leicester’s success and who actually want to go to their last home game of the season.”
Adie Wheat: “Gutted. Been all season, 1968 my first game never miss a season. Silver member can’t get a ticket for the most important game – winning the league.”
Gemma Kott: “Gold membership! Was trying to get a ticket since 9am only to be told all sold out! Never had that problem before. People selling their tickets for a massive profit should be banned from buying tickets again.”
Philip Goodchild: “Gutted. You would have thought paying £40 per season would get you a ticket. The club should have capped the memberships at the beginning of the season.”