The First Minister defended her decision, suggesting the mayor was seeking to “generate a spat” in order to position himself in a future Labour leadership contest.
It follows criticism from Mr Burnham at the weekend, who said the restriction on non-essential travel was disproportionate.
He also said his administration should have been contacted prior to the announcement.
On Friday, Ms Sturgeon said that all non-essential travel to Manchester and Salford would be banned due to rising coronavirus cases.
The First Minister was asked about Mr Burnham’s reaction on BBC News on Monday. She said: “These are public health measures.
“I have a duty, and it’s one I take very seriously, to keep Scotland as safe as possible.
“I’m sure Andy Burnham feels the same sense of duty toward people in the Greater Manchester area.
“I’ve always got on well with Andy Burnham and if he wants to have a grown-up conversation he only has to pick up the phone.
“But if, as I suspect might be the case, this is more about generating a spat with me as part of some positioning in a Labour leadership contest in future, then I’m not interested.
“We’ve all got a serious job of work to do right now and I’m serious about doing that job in a way that keeps Scotland as safe as I possibly can.”
Earlier, Deputy First Minister John Swinney also defended the decision to impose a travel ban.
Mr Swinney also said Mr Burnham’s call for compensation was not “a relevant point”.
He told the Today programme: “We have in place, in Scotland, business support that we have made available to companies to try and sustain them, there will be support in place in England for exactly the same circumstances.
“We have got to take decisions based on the data that presents itself and sometimes that is very uncomfortable data for us.
“We have to act quickly to try to make sure we are doing everything possible to suppress the spread of the virus, and that is what members of the public would expect of us.”