Mr Brown said he was given the commitment during a meeting with Transport Secretary Chris Grayling earlier this week.
The Scottish Government supports the move to build a third runway at Heathrow Airport, saying it will bring significant benefits to Scotland.
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Mr Brown said: “We have long called for Scottish airports to be given additional slots at Heathrow and welcome this commitment from the UK Government.
“This has been a key priority for the Scottish Government for some time, so we are very keen that concrete action is taken to build on this commitment and ensure that this vital additional capacity is safeguarded.
“It will be important to map out a clear process to secure a minimum level of access to Heathrow for Scottish airports which is compliant with all relevant regulations and legislation, in particular state aid regulations.
“The Scottish Government is happy to provide whatever support is necessary to achieve these aims and we look forward to further discussions on this issue in the future.”