He told MSPs that the operators of Prestwick Airport had a bid team in place.
The owners of Campbeltown airfield were also interested in attracting the venture.
Stornoway, Newquay and Llanbedr are also on a UK government shortlist of potential sites.
RAF Leuchars has been identified as a potential temporary facility.
Ministers are keen to see the spaceport established by 2018. It would be used to launch commercial flights and satellites into space.
Speaking in the Scottish Parliament, Mr Brown said the Scottish government "was keen" to see the port built in Scotland.
He said: "I am aware Glasgow Prestwick Airport is making considerable progress.
"It has a bid team in place and is working with a range of local and national partners to develop its proposal."
Mr Brown added: "The owners of Campbeltown air base are also interested in progressing a bid."
Campbeltown is a former military air base and part of the site is used by Highlands and Islands Airports Limited (Hial).
Earlier this month, the leader of the council in the Western Isles said Stornoway had "critical features" that made it the "ideal location" for a spaceport.
Angus Campbell, of Comhairle nan Eilean Siar, said Lewis's geography and access to research and development facilities were suited to a facility.
Hial, which operates Stornoway Airport, said it would potentially support a venture.
Two Scottish airports are actively bidding to host the UK's first spaceport, according to Infrastructure Secretary Keith Brown.