The two clubs had been in jeopardy heading into the final round of fixtures on Sunday but ultimately Everton’s 1-0 victory over Bournemouth ensured they avoided the drop.
At one stage it looked as though it could be the Foxes who stayed up as they took an early lead against West Ham but their eventual 2-1 win was rendered academic by Everton’s result.
The Toffees ended the season in 17th place, two points ahead of Leicester, while Leeds were also relegated after a 4-1 loss to Tottenham.
Former England striker Lineker, who began his career at Leicester before spending a season at Everton in the mid-1980s, tweeted: “Absolutely gutted, but glad it’s Everton. Have a lot of love for that great football club. Congratulations.”
Leicester’s relegation comes seven years after they were crowned Premier League champions and just two years after they won the FA Cup.
Lineker added: “A word on Leicester. If eight years ago, you’d have given me the option of winning the Premier League and the FA Cup and then get relegated, I’d have snapped your hand off. Also I’d have told you not to be so utterly ridiculous.”
It has been a dismal season for Leicester and TV pundit Roy Keane was not sure how quickly they could recover.
The former Manchester United midfielder said on Sky Sports: “They didn’t seem to get any momentum into the season from a bad start. It’s no surprise to see them where they are.
“Clubs can bounce back but it isn’t easy. I think it is a rebuilding job at Leicester.”
Leeds’ three-year stint in the Premier League ended in a whimper as they were thrashed by Spurs at Elland Road.
The club had brought in Sam Allardyce in a last-ditch attempt to escape relegation with four games remaining but the former England boss was unable to engineer a recovery.
The team collected just one point from Allardyce’s games and finished in 19th position, five points behind Everton.
Keane was scathing of their performances.
He said: “They’ve looked weak over the last month or two, even with Sam coming in.
“They were fighting for their lives today and conceded four goals at home. That’s nowhere near good enough.
“Sam obviously came in too late. Defensively they look so weak. Some of the goals – it’s almost pub team defending.
“Not strong enough mentally, that desire – nowhere near good enough.”
Chelsea, meanwhile, aimed a parting shot at Leeds on social media.
Rivalry between those two clubs dates back to some hard-fought clashes in the 1960s and 70s.
In August, Leeds trolled Chelsea on Twitter during their 3-0 victory over the London club.
In that game, Chelsea tweeted the Blues were “starting to assert ourselves” just moments before Leeds opened the scoring and quickly followed with a second goal.
“Life comes at you fast!” Leeds tweeted in reply.
Now, nine months later, Chelsea have got their own back.
“It certainly does,” they tweeted.