Trump had told voters in the run-up to the US election that he would appoint Supreme Court justices who were pro-life.
In an interview with 60 Minutes on Sunday, Trump reiterated his pledge to allow states to decide their own abortion laws.
Donald Trump has pledged to change abortion laws in the US
Speaking to local radio station Boston Herald, mayor Martin Walsh suggested that the idea of women having to travel to other states to get abortions was "absurd".
Walsh added that he would do everything he could to protect abortion laws in Massachusetts.
He said his plans involved "working with the legislature to make sure the laws don't get weakened. Working with health centres to make sure that, if there's threats of cutting back federal funding to those health centres for any type of reproductive rights, working with all the health centres in the city of Boston to make sure there is an open-door policy.
"We have it now, but whatever we can do, making sure our public health commissioner is on top of the issue as well."
The Democrat mayor suggested that Boston could provide a safe haven for legal abortion clinics.
"I think a lot of the clinics would probably look to cities like Boston to come to, safe cities," Walsh said, "Cities that are open and progressive and understand the importance of having access."
Walsh is a practicing Catholic and was one of 35 mayors to meet with Pope Francis at a meeting about human trafficking during his US visit in 2015.