Maradona's spokesman announced the footballer had died at the age of 60 and President Alberto Fernandez immediately announced three days of national mourning in the South American country.
The player was the inspiration behind Argentina's 1986 World Cup triumph.
However, he dubbed the first of his two goals against England in the quarter-final the "Hand of God" after he punched the ball over goalkeeper Peter Shilton.
Lineker was a member of that side and also scored in the match as Argentina progressed 2-1.
"Reports from Argentina that Diego Armando Maradona has died," he tweeted. "By some distance the best player of my generation and arguably the greatest of all time.
"After a blessed but troubled life, hopefully he'll finally find some comfort in the hands of God. #RipDiego"
Some of Lineker's seven million followers took him to task for the tweet.
"Not the time to make puns," tweeted one to which Lineker replied: "Don't be ridiculous. It's heartfelt."