The Football Association is on the hunt for a new England manager after Southgate announced his decision to end his eight-year spell in charge, two days after defeat to Spain in Berlin.
Former England striker Lineker described Southgate as "tactically inept" and England's display as "s***" after their group-stage draw with Denmark.
The BBC Match of the Day presenter received plenty of flak himself for his comments, but denied criticism of England's performances and tactics by him and other pundits was responsible for Southgate's departure.
Television presenter Gary Linker, a former England forward, has been critical of Gareth Southgate. (AP PHOTO)
Lineker said on The Rest Is Football podcast: "We (pundits) were critical at times, because you can't say when a team is not playing very well, that they played well.
"If they played awfully, you say they played awfully, and they did in the early part of the tournament.
"There's no question about that. But then to try and make it that that's the reason, that criticism is the reason he's gone. It's not. It's not that at all.
"We speak to the players and they understand where it's coming from."
Jude Bellingham described Southgate as "easily one of the best coaches in the history of the national team" when leading the players' tributes on Tuesday and said he was "an unbelievable human being".
Southgate failed to deliver England's first major trophy since 1966, but led them to two European Championship finals and the semi-finals of the 2018 World Cup. In the modern era, only four predecessors have had a better win percentage (59.8 per cent).
Lineker, who scored 48 goals in 80 England appearances, added: "I think he's been the right person to bring the nation together in terms of the football team.
"And, now, maybe it's time for someone else with a more modern, attacking style of football. Because I think the game's gone away from being successful if you're really defensive."
England Under-21s boss Lee Carsley, Pep Guardiola, Eddie Howe, Mauricio Pochettino, Jurgen Klopp, Graham Potter and Thomas Tuchel have all been linked with the England vacancy following Southgate's departure.
Newcastle boss Howe and former Brighton and Chelsea manager Potter currently top the bookmakers' lists.
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