Airlines UK, the trade body for UK registered airlines, has written to the chancellor Rishi Sunak seeking an extension, with a slow recovery for air travel forecast.
EasyJet and British Airways are among the carriers to have capitalised on the scheme so far, with BA having furloughed in excess of 30,000 staff.
In its letter, Airlines UK warned a return to normality for international aviation would be "gradual", and that aviation was facing a "cliff-edge post-June" once the furlough scheme lapses on 30 June.
The group said the government should consider reviewing the scheme on a sector-by-sector basis as some parts of the economy recover as a faster pace than others.
Iata has stressed recovery in air travel will be slow with European air traffic currently down 90%.
Elsewhere, the BBC reports an all-party group of 20 MPs is seeking a government bailout for aviation, underpinned by stringent environmental commitments and conditions.
The Unite union, meanwhile, has set out a "radical" blueprint for the future of aviation, in which it says government should be able to inject cash into airlines to preserve vital connectivity, both for travel and economic prosperity, and to save jobs.
Like the MPs calling for a bailout, the measures would require commitments on the environment and corporate practices.
Earlier this month, easyJet became the first UK airline to tap the government’s Covid Corporate Financing Facility, to the tune of £600 million, while Wizz Air has confirmed it is eligible to draw cash from the scheme.
Jet2.com parent Dart Group is currently seeking to establish its eligibility.
Virgin Atlantic is seeking state support, but it is unclear at this stage what avenue it is pursuing. The government is understood to have knocked back its first application.
Sunak has previously warned airlines to look for assistance from its existing shareholders and the private sector before approaching government.
