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The travel industry groups want the remaining Covid restrictions on travellers to be removed.
According to data last week, one in 25 people in the UK have the virus, but airlines say passenger testing is not having any impact.
The government said it was keeping all measures under review.
Testing for UK arrivals and departures has held back the sector's recovery, according to the travel industry.
The prime minister will urge his cabinet to back his decision not to impose any more Covid restrictions in England.
Boris Johnson said on Tuesday that he hoped the country could "ride out" the current wave, although he acknowledged that parts of the health service would feel overwhelmed.
If you're travelling to the UK for the first time, you have to take a negative test, which can be a PCR or a lateral flow test, up to two days before departure.
They have to take another test within the first two days after arriving in the UK.
The number of new cases reported in the UK each day was between 40,000 and 50,000 at the time of the rule being brought in, but it was only rising slowly because it was almost entirely made up of the Delta variant of Covid.
The UK has announced more than 200,000 new cases in a day for the first time in the entire Pandemic, so airlines can argue that there is no hope of keeping Omicron out.
The US and France are the only countries that have had more than 200,000 positive tests in a single day.
The current measures would be financially disastrous for the industry according to Airlines UK. The current limits have depressed demand.
Manchester Airports Group claims that the research it commissioned shows that pre-departure testing has had little or no impact on the spread of Omicron.
The passenger numbers at the airports fell after Omicron measures were introduced.
If Omicron became the dominant variant, there would be no need for travel restrictions, said Health Secretary Sajid Javid last month.
Future plans for domestic testing must also be considered.
The system has come under pressure in recent weeks as cases have surged, with appointment slots and home test kits temporarily unavailable.
100,000 critical workers will be required to take daily tests.
According to the Daily Telegraph, the government is planning to change testing rules so that people who test positive for a drug don't have to take a confirmatory test.
The government pointed to the UK Health Security Agency's guidance that says people should take a follow-up test if they get a positive result.
The agency said that they continue to review and make more slots available for booking.
Private tests must be paid for by travellers.
The government should remove travel testing requirements which come at a huge cost to the industry, according to the chiefs of the two companies.
They wrote in a statement that it was vital they did not stay in place a day longer than was necessary.
The temporary testing requirements were introduced to prevent Omicron cases from entering the UK and stop people from passing it on to others, according to a government spokesman.
The government is likely to drop the requirement for people to take a pre-departure test before they arrive in England, according to The Times. The requirement that people take a test within two days of arriving in England is still in place.
The European airline and tour operator stocks rose on Tuesday.
"With so many people being forced to isolation at home, it's likely many will be spending the next few weeks browsing travel blogs for inspiration, given there is so much desperation for a holiday," said Streeter.
A recent report by aviation analytic firm Cirium found that the Covid pandemic caused a 71% drop in international flights in and out of the UK in 2021.
Travel.
Omicron variant.
A coronaviruses epidemic.
