Lampard has been out of work since returning to Chelsea in an interim capacity at the end of the 2022/23 season, but had worked with Southgate and the national team in the build-up to the Euros.
"This will be Gareth Southgate’s last tournament," Lineker prophecised on The Rest Is Football podcast. "I think, whatever happens. If he wins, I think he’ll absolutely bow out, and if he doesn’t then I think he’ll bow out. It’ll be his decision.
"But I wouldn’t disregard Frank Lampard. I think he tactically gets it. I think the players will respect him immediately. I’m getting ahead of things here. If you’re going to go English, you go… who else? Eddie Howe? Eddie Howe, possibly?
"I think he’s (Lampard) been a bit unlucky in his managerial career in some ways. I think he gets it tactically."
Frank Lampard alongside Gareth Southgate during an England training camp. (Image: Getty)
Lineker's podcast co-host Micah Richards was similarly enthused by the prospect of Lampard leading the national team, after being blown away with his analysis during Euro 2024 for the BBC.
"Frank’s analysis was about three minutes. And, normally, if someone goes that deep, after maybe two minutes, you’ve lost interest," the former Manchester City defender explained. "But I was glued to it for three minutes. And you can tell he’s just got something.
"He’s articulate. The way he explains points. You can tell there’s a manager there inside of him. I’ve really enjoyed working with him."
Lineker then echoed his sentiments: "That three minutes Frank did was amazing."
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"Hopefully something comes up that feels the right thing for me, I'm keen to get working again but as I reiterate, I am enjoying the [family life]," Lampard told talkSPORT. "It's an intense job so when you're out of it, it's nice to appreciate being around all my children, my young children particularly and spending more time with those."
When asked if the intensity of management was offputting, the 106-cap England icon replied: "No, not at all. You love the job, you understand the rigours of it in the modern day, we see it so much.
"I've enjoyed all the clubs I've worked with, they've all been big challenges obviously for different reasons. I enjoy trying to work with players, improve players, improve teams, so let's see what comes."