This comes as the Prime Minister unveils new immigration rules to make it easier for leading scientists to live and work in the UK after Brexit.
There are reports Mr Johnson could seek to hang on long enough to ensure Britain is out of the EU before going to the polls if he is defeated in a vote of confidence when MPs return next month from summer recess.
If Mr Johnson is defeated in a no-confidence motion he would have 14 days, under the Fixed Term Parliaments Act, to win another vote of confidence or, if no other government could be formed, face a general election.
In his letter to Sir Mark, Mr Corbyn said such a course of action would be "unprecedented" and "unconstitutional".
He asked Sir Mark to confirm that if the UK is due to leave the EU without a deal while an election is underway, the Government should seek another time-limited extension to Article 50 to allow the voters to decide.
He wrote: "Forcing through no-deal against a decision of Parliament, and denying the choice to the voters in a general election already under way, would be an unprecedented, unconstitutional and anti-democratic abuse of power by a Prime Minister elected not by the public but by a small number of unrepresentative Conservative Party members."
But Brexiteers argue Parliament has already voted to leave the EU, voting to trigger the Article 50 process and passing legislation setting Britain's withdrawal date for 31 October.
A senior Conservative source said: "Jeremy Corbyn will do anything to get his hand on the keys to Number 10.
"No amount of letter-writing political stunts will change the fact that politicians don't get to choose which public votes they respect."
The Prime Minister has also sought to address concerns over access to the UK after Brexit, announcing a new fast-track visa route which will be created for foreign scientists coming to the UK.
Unveiling the plans on a visit to the Culham Science Centre in Oxfordshire on Thursday, Mr Johnson said he wanted to ensure the country's immigration system attracted "the very best minds from around the world".
He said: "Britain has a proud history of innovation, with home-grown inventions spanning from the humble bicycle to the lightbulb.
"We were home to the world's first national DNA database, we discovered graphene, and our cutting-edge scientists should be proud to follow in the footsteps of titans like Ada Lovelace and Nobel Laureates Francis Crick and Peter Higgs.
"But to ensure we continue to lead the way in the advancement of knowledge, we have to not only support the talent that we already have here, but also ensure our immigration system attracts the very best minds from around the world."
The fast-track immigration route will be designed to attract researchers and specialists working in science, engineering and technology.
Mr Johnson added: "I want the UK to continue to be a global science superpower, and when we leave the EU we will support science and research and ensure that, far from losing out, the scientific community has a huge opportunity to develop and export our innovation around the world."
However, the prime minister's proposal has been dismissed by the world-leading physicist Professor Sir Andre Geim, whose discovery of graphene was praised by Mr Johnson.
Sir Andre and other prominent British scientists have warned that leaving the EU, especially via a no-deal Brexit, would damage the UK's status as a world leader in science.
Sir Andre told The Times: "The government may try and reduce the barriers to entry for scientists but they cannot reduce turmoil that would be caused to science in the UK by a no-deal Brexit."
"Scientists are not fools. They know that turmoil is inevitable for many years."
Sir Paul Nurse, director of the Francis Crick Institute, warned that British institutions would face barriers in participating in international collaborations after Brexit.
He told the paper the benefits of participating in Europe reach beyond the economic.
Sir Paul added: "Ninety-seven per cent of research groups at the Francis Crick Institute would prefer to stay within the EU's Horizon Europe funding programme, as it promotes meaningful collaboration and has a reputation that no domestic scheme can hope to replicate."