More than 31,000 people have been flown back to the UK from 27 countries under a charter flight scheme launched to rescue people stuck abroad due to the coronavirus lockdown.
But Foreign Office Minister Lord Ahmad said the programme would be "paused" after its current phase, which ended last week.
MPs and peers have warned it could leave an estimated 300,000 British people stranded abroad, many of whom are believed to be in India.
More Covid-19 coverage:
Black Country MPs including Stuart Anderson and Pat McFadden say they have been inundated with calls and emails from family members of stranded constituents.
Many of them with 'Indefinite Leave to Remain' status – meaning they live and work in the UK but do not have British passports – say they have not been prioritised in the scheme.
Wolverhampton South East MP Pat McFadden said: "It has been a huge logistical exercise to bring home British residents stranded in India but the Government must not stop before the job is complete.
"People with Indefinite Leave to Remain are not temporary visitors to this country. The UK is their home. Many of them have lived here for decades and have worked hard and paid their taxes the same as anyone else.
"They can’t just be abandoned abroad while flight restrictions remain in place. I think the Government will be making a huge and very damaging mistake if it ends this operation while sizeable numbers of UK residents remain stuck in India."
The issue has been raised with the Foreign Office by Warley MP John Spellar and Wolverhampton South West MP Mr Anderson, who have called for all stranded Brits to be given the same priority.
The Government has chartered around 150 flights back to the UK, including 64 from India, with the most recent five flights taking off from the cities of Amritsar and Ahmedabad between May 12 and 15.
Lord Ahmad said 16,500 Brits have been brought back from India. "We shall pause the charter programme after the current phase, though we will keep this decision under review," he added.
Lib Dem former leader Lord Campbell of Pittenween has suggested that around 300,000 UK citizens still needed to be repatriated.