The move comes five months after a sale to an undisclosed bidder fell through, which The Scotsman reported was Glasgow Airport owner AGS.
It is thought French firm Vinci, which runs Gatwick and Belfast International airports, may be interested in Prestwick but the company declined to comment about whether it had bid.
Ministers put the South Ayrshire airport up for sale in 2019 after buying it for a nominal £1 to avert closure by its owner Infratil and hundreds of job losses.
The Scottish Government declined to say how many offers it had received.
It is not clear if the sale will include the repayment of some £40 million in Scottish Government loans.
Mr Matheson said: “Since the Scottish Government bought Glasgow Prestwick Airport (GPA) in 2013, we have been clear that it is our intention to return the business to the private sector when the time is right.
"In November, a notice was placed in the Official Journal of the European Union inviting expressions of interest.
"That attracted credible interests and since then GPA and its professional advisers have carried out a detailed evaluation of the competing bids.
"I am pleased to report significant progress has been made and I have agreed the selection of a preferred bidder.
"This allows the process to move to the next stage and potentially an early divestment of our shareholding.
“To protect the integrity of the process, we will not reveal the identity of the preferred bidder at this stage.”
Mr Matheson said last September when announcing the previous sale had fallen through: “Covid-19 has impacted on the aviation sector globally.
“We have been advised the company selected as preferred bidder does not wish to complete the purchase at this time.”
Scottish Labour transport spokesperson Colin Smyth said: “It is vital any deal to sell Prestwick Airport must include a clear commitment from the buyer to retain and grow jobs at the airport, which are vital to the economy of Ayrshire.
“Any deal should also have a clear plan showing how the taxpayer will get back the tens of millions of pounds loaned to the airport from the public purse.
“We must not allow a cut-price sale for the sake of the SNP Government washing its hands of the airport unless it gives the taxpayer value for money.”
Graham Simpson, his Scottish Conservatives counterpart, said: “This is positive news and the hope is that it will not prove another false dawn.
“With restrictions set to continue for some time to come on the aviation sector, the need for SNP ministers to conclude a successful sale is urgent.
“Too much taxpayers money has been lost as a result of the SNP’s mismanagement of Prestwick Airport.
"We must hear if those millions are going to be recovered as part of the sale process.”
Scottish Liberal Democrats transport spokesperson Carole Ford said: "I hope the new owners can rejuvenate the airport and ensure it escapes its controversial reliance on the US military for custom.
"Having bought the airport for a pound, it is not good enough for ministers to sell for £2 and call that a profit.
"Prestwick airport has been kept alive with more than £43m of taxpayers’ cash.
“The transport secretary needs to be transparent about how much of that investment will be repaid.”