The group’s chairman, Sir Gerald Howarth, has written to the Prime Minister asking her to end the ban and claimed that significant progress has been made on tightening security checks at the Red Sea resort’s airport.
He also warned that there was a risk of people in Sharm becoming “radicalised” if the area’s tourism industry continued to suffer.
Flights between the UK and Sharm were suspended by the Government in November 2015 after the terrorist bombing of a Russian passenger plane, killing 224 people.
Britons at the resort were brought home in a series of rescue flights with extra airport security.
During a visit to Egypt in July Sir Gerald inspected how baggage was handled at the airport and met senior officials.
The Conservative MP for Aldershot said: “They really have been listening to what the UK authorities have been asking them to do. They are clearly bending over backwards to try to deliver.
“One of the guys we met was a Department for Transport (DfT) official and when I asked him, ‘Are there any showstoppers to the resumption of flights?’, he said ‘No’.
“I believe that the necessary measures have been put in place to minimise the risk.
“We live in an imperfect world and no government can guarantee 100% security anywhere.”
In June, British Airways suspended its flights to Sharm “indefinitely”, stating that “the safety and security of our customers will always be our top priorities”.
EasyJet, Monarch and Thomson Airways have previously said they are awaiting for the Foreign and Commonwealth to change its travel advice, which currently warns against all but essential travel by air to or from the resort.
Sir Gerald explained that hotels in Sharm were only 25-35% occupied and were “clearly hurting” from the UK flight ban.
“They’ve invested a huge amount of money in these really very sophisticated places,” he said.
“They’ve kept on the staff. They can’t do that indefinitely and, if the staff are laid off, we will have a substantial number of people who will be potentially radicalised.
“There is an imperative on ensuring that the resort can resume its contribution to the prosperity of Egypt.”
Last month, May welcomed Egypt’s efforts to improve security at Sharm airport in a phone conversation with the country’s president Abdel Fattah el-Sisi.
An account of the call released by Downing Street gave no indication the Prime Minister was ready to ease Government advice, saying only that the UK would continue “working closely” with Cairo on the issue.
A Downing Street spokesman said: “The Prime Minister said she recognised the economic effects of the suspension, given the importance of the tourism industry to Egypt’s economy, and she praised the Egyptian government’s ongoing efforts to improve security at Sharm el-Sheikh airport.”
Sir Gerald said the resumption of flights was “exclusively in the hands of the Prime Minister” and he believed “she is keen to make a decision in the next few weeks”.
He added: “We hope that we can resolve this problem.”
A DfT spokeswoman said: “The security of British nationals is our top priority, and we took the decision to suspend flights from Sharm el-Sheikh in November last year to protect the travelling public.
“We keep aviation security under constant review.”