The full match and highlights can currently be watched on BBC iPlayer but will expire on Thursday.
Scotland beat Denmark 4-2 at Hampden Park last month to reach the World Cup finals for the first time in 28 years.
Paul McLennan, the SNP MSP for East Lothian, previously raised the issue in a debate in the Scottish Parliament.
Speaking on Monday, he said: “I am sure I’m not the only Scotland fan who would love to re-live our incredible final qualifying match over coming months as we look forward to next summer.
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“Wouldn’t it be great if the BBC could make the match available for all to watch again right up until the tournament kicks off in June?”
As there is currently no deal for future games, the SNP said there are concerns that there could be a return to the situation where Scotland matches are only available to watch on expensive pay-per-view channels.
McLennan expressed this worry, and urged football authorities and broadcasters to come to an agreement to ensure that matches are available for all to watch on free-to-air TV.
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“I urge the footballing authorities, broadcasters and the UK government, which has control over broadcasting, to come together to ensure that all future Scotland men’s and women’s football matches can be enjoyed by everyone – and that a whole next generation can be inspired to take up sport as a result,” he said.
If successful, the call would mean that the qualifying match would be available until Scotland play their first game against Haiti at 2am on June 14, 2026, UK time.