Mrs Foster wrote the letter in 2015 when she was the Finance Minister.
She told the BBC last week she had no recollection of the letter to then local government minister Marco Biagi.
No correspondence had been sent in a personal capacity, she added.
At the time the Scottish government was bringing in legislation to allow a civil partnership made in another jurisdiction to be recognised in Scottish law as a marriage.
When asked about the letter by the BBC, Mrs Foster said: "I'm not quite sure what he (Mr Biagi) was referring to but it certainly wasn't a letter from me and I've no recollection of a letter from me.
"If I'd written to him officially as Minister of Finance or something like that around recognition laws here in Northern Ireland, I have no recollection of it. I certainly didn't write in a personal capacity."
Mrs Foster's letter came six months after her party colleague and predecessor Simon Hamilton wrote to Mr Biagi.
That letter has also been released, along with Mr Biagi's replies, following a BBC News NI Freedom of Information request.
The letters show that Simon Hamilton asked the Scottish government in March 2015 either to not exercise a power to convert non-Scottish partnerships or to exercise the power "in such a way as to exclude Northern Ireland civil partnerships".
Mr Biagi wrote back to Mr Hamilton stating that it was the Scottish government's intention to allow a civil partnership made in a jurisdiction outside Scotland to become a same-sex marriage.
Then in September 2015 Arlene Foster, who had succeeded Mr Hamilton as finance minister, wrote to say she shared her predecessor's concerns about the proposal.
She also argued that civil partnerships formed in Northern Ireland should be excluded from being allowed to become same-sex marriages in Scotland.
"In this instance we can achieve legal certainty by restricting the definition of a 'qualifying civil partnership' so as to exclude civil partnerships which were entered into in Northern Ireland," she wrote.
A letter has been released showing that Arlene Foster wrote to the Scottish government to say allowing NI civil partnerships to be converted to Scottish same-sex marriages would mean legal "uncertainty".