Last week, the First Minister said all non-essential travel to Manchester and Salford would be banned due to rising coronavirus cases.
Burnham, the mayor of Greater Manchester, hit out at Ms Sturgeon, claiming he was not told about the ban, which is now in effect, prior to the announcement.
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.
Ms Sturgeon was asked about Mr Burnham’s reaction on BBC News on Monday.
She said: “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.”
On Wednesday, the pair held a meeting in a bid to resolve the issue – but Burnham has since accused Sturgeon’s camp of leaking confidential details.
A spokesperson for Burnham said: “We are surprised that the Scottish Government has briefed details of a discussion in what we were told was a confidential meeting.
“But, given this has happened, we need to be clear on what was said.
“The First Minister accepted that she should have told the Mayor before she announced her decision.
“We also welcome her acceptance that the Scottish Government needs to provide more information about how they reach their decisions on bringing in regulations and travel bans that affect other parts of the United Kingdom.
“This is important because towns and cities in England need to know if they may be put under Scottish regulations.
“We are, however, disappointed that the First Minister completely dismissed the financial impact her decision has had on residents in Bolton, Manchester and Salford and the need for her Government to consider providing compensation.
“These people had no advance notice and no ability to change their plans and are now left out of pocket.