Responding to an article about the news, Lineker tweeted: “I think only Twitter can take people off Twitter.”
It comes after director-general Tim Davie said the BBC is to publish new guidelines under which it could suspend employees’ Twitter accounts.
Mr Davie said the rules would cover those working in news, current affairs and beyond and were “imminent”.
He was appearing before the Digital, Culture, Media and Sport (DCMS) select committee to answer questions on a number of issues facing the BBC, including its cost-cutting plans and the likely impact of the proposed decriminalisation of the non-payment of the licence fee.
Mr Davie, who took over the role of BBC director-general from Tony Hall, said: “We are going to be publishing in the next few weeks, and this is imminent, clear social media guidelines, and they will cover both news and current affairs, and beyond news and current affairs.
“We will have, within those guidelines, the enforcement policies will be very clear.
“We will be able to take disciplinary action. We will be able to take people off Twitter. I know people want to see hard action on this.”
“I am prepared to take the appropriate disciplinary action, all the way to termination,” Mr Davie said.
When asked for further details on how this would be enforced, the BBC said only that the guidelines would be released in the coming weeks.
He also commented on sports presenter and pundit Lineker, who it was recently revealed has just signed a new five-year contract with the broadcaster, with a 23 per cent pay cut – from £1.75 million to around £1.35 million.
Lineker is known to be outspoken on social media and there has been media speculation that he might not be forced to adhere to the new rules at the corporation.
The sports presenter also previously appeared to shrug off suggestions that he should be "terrified" of Mr Davie's new rules, replying "nah" to a post about the clampdown.
— Gary Lineker (@GaryLineker)
Addressing the issue, Mr Davie said: “I would note that Gary Lineker has actually been very clear in his statements recently, saying ‘I understand I have responsibilities when working at the BBC’.
“Those responsibilities will be clearly laid out and my belief is, as I say, I am now the director-general so I am running the show, and in my view party political statements are not the right thing for people to be making if they are, as part of an impartial news organisation.
“I mean, we will come back with social media guidelines to make clear where the lines are.
“If someone is a face of the BBC I think entering into party politics seems to be not the right place to be and I’ve been very clear about that.”
Mr Davie later added: “The good thing is I will be making that even clearer as I go through my social media guidelines.”
The BBC has warned that switching to a civil system for the licence fee would cost the broadcaster more than £200 million a year.
Addressing the proposed decriminalisation of non-payment of the licence fee, Mr Davie said it was “a bit early for us to make assumptions about what the Government will or won’t do”.
He said the BBC’s position was “crystal clear on this”, adding that “this system is logically hard to beat”.
He said: “If you are a law-abiding, paying customer and licence fee payer, what I am trying to do is get you the most value for money, and the decriminalisation proposal, frankly in my mind, just doesn’t pass the logic test.”
After taking over the job, Mr Davie said the BBC cannot be “complacent” about its future and must “renew” its commitment to impartiality.
Addressing a question on the topic, Mr Davie said his “key concerns were not historical issues in terms of left or right politics” but were focussed on “audiences and audience values”.
Mr Davie said it was critical the BBC was not associated with “one particular point of view or perspective on life”.
Mr Davie succeeded Lord Tony Hall as director-general at the beginning of the month.
He was acting director-general for four months following George Entwistle’s resignation in November 2012, before Lord Hall’s appointment, and previously served as the chief executive of BBC Studios.