Mr Mackay, who resigned over messages he sent to a teenage boy in 2020, was the minister responsible for the sign-off of the contract for hulls 801 and 802 in 2015.
The two ships are now running five years late and more than £150 million over budget.
The public audit committee in Holyrood is continuing to scrutinise elements of the decision to award the contract to Ferguson Marine, then owned by tycoon Jim McColl, without the standard full builder’s refund guarantee, which would have protected the taxpayer following delays or overspending.
Ministers have repeatedly defended the deal and the later decision to nationalise the company in 2019, saying it saved the yard from closure and saved hundreds of jobs.
However, Audit Scotland has criticised the Government for not adequately recording why ministers believed the risks were acceptable, with Mr Mackay expected to answer questions on this matter.
It is understood the committee has not ruled out also inviting the former minister to give oral evidence in Parliament following his written evidence.
Mr McColl, whose feud with the Scottish Government resulted in him labelling Nicola Sturgeon a liar in recent weeks, has also been invited to give evidence in person.
Erik Østergaard, the former chair of Caledonian Maritime Assets Limited (CMAL) – the ferries infrastructure public body – is another invited to give evidence around his warnings to ministers about the deal.
Committee convenor Richard Leonard said Transport Scotland and Scottish Government officials would give evidence next week.