Viewers were left amused after the Health Secretary went "all bashful" after being asked about the rules by Ms Burley, who herself question why she was "whispering".
She then told Mr Hancock, "it's ok to smile", as he struggled to get his words out in response.
Asked about the Government’s guidance that only “established” couples should be having sex, he told Sky News: “In these rules that we have to bring in, there have to be boundaries, to coin a phrase.
“If you’re saying that two households shouldn’t mix, which we are in some parts of the country – in the North East, the North West, in Scotland, in parts of Wales – then you have to then define what is the boundary of that.”
Ms Burley pressed him on what constitutes an established relationship, saying "some people say [I love you] and don't mean it."
Mr Hancock said: “I think we should stick to the letter of it, which is it is okay in an established relationship."
“It just means that people need to be careful, they need to be sensible.
“If you’re in a relationship that is well established… what it means is people realising that coming into close contact with people from other households, then that is how the virus spreads.”
Mr Hancock joked that “I know I am in an established relationship” with his wife.
Viewers saw the funny side of the exchange, with some joking that Mr Hancock and Ms Burley were getting on well.
So funny, he went all bashful 😂— Robbie Williams (@robbie9123)September 24, 2020
One person watching said: "I think he should buy you dinner first". Another added: "So funny, he went all bashful."
I think he should buy you dinner first 😎— Anthony Beck (@AnthonySBeck)September 24, 2020
A third person said: "Can’t believe you asked our health secretary about casual sex."