The Match of the Day host last week signed a letter criticising the government’s Rwanda scheme.
That led to a backlash from some Tory MPs, which then prompted Lineker to attack Grant Shapps, Jonathan Gullis and Lee Anderson on social media.
In an article in the Sunday Telegraph, Dowden hit out at “amateur BBC pundits offering as much insight as I could on football tactics”.
On Sunday Morning with Trevor Phillips on Sky News this morning, the presenter asked the deputy PM: “Do you want to say who you had in mind then?”
Dowden replied: “I think it’s fairly obvious some of the people who have been spewing their views forth on this.”
Phillips then said: “Could the words ‘Gary Lineker’ be floating through this conversation?”
The deputy PM said: “I think people should swim in their own lane. I know nothing about football punditry, I can assure you about that.
“The wider point I was making is there’s a lot of talking this country down. Actually, I see a country with a manufacturing sector larger than France’s, we’re dominating in artificial intelligence.
“There’s so many reasons to be optimistic about this country, and this doomster scenario that they’re constantly predicting doesn’t actually happen because there’s real strength in our economy.”
Phillip then asked: “So the message to my fellow anchor Gary Lineker is stay in your lane?”
Dowden replied: “Each person should play to their strengths.”
He said: “The BBC’s reputation matters and this isn’t helpful so we do need to find a solution to it and, were I to be chair, I’d be keen to try to bring about such a solution.”