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.

Rob Stewart was speaking after plans were unveiled for 3,500-seat arena, an aquarium and a city beach area with bars, restaurants, hotels and homes.
It will be the biggest landscape change since the city was blitzed during World War Two and is expected to lead to about 1,700 permanent jobs.
Swansea council unveiled the plans on Friday with work due to start in 2017.
It comes 10 years after a flagship scheme to redevelop Swansea city centre collapsed.
Mr Stewart said: "We've been very careful to make sure what we've proposed is deliverable. We're very confident we have the right developers and right funders."
Two developers - Rivington Land and Acme and Trebor Developments - have been appointed and construction work will bring about 500  jobs.
The main areas to be worked on are the former St David's shopping centre, land next to the leisure centre and the civic centre site.
Mr Stewart added: "People quite rightly would want to see us get on with this as soon as possible.
"We are already in discussions with Welsh government, with the UK department of trade and industry in London, to make sure we can take these plans forward as quickly as possible."
It will run from Whitewalls, include a cafe quarter and be built on an incline so that at the other end, people can walk straight over Oystermouth Road to where the arena will be, hosting concerts, shows, conferences and ice skating.
There will also be a high-rise building next to it comprising of a hotel and flats and could become the tallest in Wales.
An access route will run from there to the aquarium and city beach.
Two Scottish airports are actively bidding to host the UK's first spaceport, according to Infrastructure Secretary Keith Brown.
[NEXT_CONCEPT]
A new £500m plan to transform Swansea city centre will succeed, the leader of the city's council has said.