The denial comes after the Mail on Sunday reported that the former British prime minister met the president's son-in-law and key adviser, Jared Kushner, at the White House last week.
The paper claims the meetings were to discuss working for Mr Trump.
Mr Blair's office said the story was an "invention" and that he continues to work for peace "in a private capacity".
After he left Downing Street in 2007, Mr Blair took the role of Middle East envoy for the Quartet Group, made up of the European Union, US, Russia and United Nations.
His spokesman said: "Mr Blair has made no such 'pitch' to be the president's Middle East envoy.
"Neither has he had any discussions about taking such a role or any role working for the new president. He has been working on the peace process for 10 years.
"He continues to do so. He does so in a private capacity. He will continue to do it in that way. Period."
Mr Blair launched a campaign last month to try to persuade the British people to change their minds about leaving the European Union, following the Brexit referendum held last year.

Unite said 1,000 of its members would walk out from 02:00 BST next Tuesday.
The union accused First of taking an "extremely aggressive" stance by sacking a branch secretary and two other workers for "legitimate trade union activity".
First Leeds said the "unjustified action" would affect tens of thousands of people across the city.
Drivers have already held two 24-hour strikes, with Unite claiming they earn Â£2 less an hour than other First drivers in the region.
The firm dismissed this as "nonsense" and said the union had failed to back up "this ridiculous suggestion".
First Leeds also disputed the "aggressive" stance alleged by Unite and said it was investigating some very serious allegations.
No members of staff have been disciplined for taking part in industrial action or for any legitimate trade union activity, it said.
First Leeds offered a 38p per hour rise frozen until May 2018, which Unite described as "insulting".
Unite regional officer Phil Bown said: "The management needs to stop ducking and diving and commit to holding genuine and meaningful talks, otherwise this strike could go on for some time with an adverse impact for bus users in Leeds - something we have desperately sought to avoid."
Paul Matthews, managing director of First Leeds, said: "I'm very disappointed and deeply frustrated that Unite plan to cause such huge disruption for our customers in Leeds.
"This is not a last resort as Unite claim - not only do we have a planned meeting before Tuesday to try and resolve the dispute, but Unite have repeatedly refused to let all staff vote on our above inflation offers."
The bus provider plans to provide a reduced service in Leeds, with buses in Bradford, Huddersfield and Halifax all unaffected.

The UK's top employers recruited more graduates this year than they expected, says the High Fliers report, based on a poll of 18,000 final-year students.
The increase is the biggest for four years and follows an unprecedented 23.3% dip between 2007 and 2009.
But graduate numbers have increased by 85,000 to 365,000 since 2007.
This suggests a further increase in graduate-level jobs will be required to keep pace with the higher number of graduates looking for work.
The report says the average number of entry-level graduate roles on offer at the UK's top companies is now 190 per employer.
And a third of these companies increased graduate recruitment targets for 2014.
Overall a rise of 9% had been predicted, compared with the actual rise of 11.6% this year.
The highest number of new graduate jobs are in the top accounting and professional services companies.
This sector saw a 23.8% increase in graduate intake.
The report says: "These latest results mark a very welcome improvement in the graduate job market and suggest that graduate recruitment at the leading employers is now at its highest level for seven years."
Although competition for graduate jobs has eased slightly, it remains tough, with an average of 39 applications per vacancy.
Four-fifths of the UK's leading employers require at least an upper-second-class degree.
And employers received at least 50 applications per position for graduate jobs at:
But more than two-thirds of companies plan to match or expand their recruitment levels next year.
High Fliers managing director, Martin Birchall, said: "It's great news for the 'Class of 2014' leaving university this summer that Britain's top employers have stepped up their graduate recruitment so significantly this year."
Tony Blair has not pitched to become a Middle East envoy for US President Donald Trump, a spokesman has said.
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Bus drivers and other workers for First Leeds are to stage an indefinite strike in a worsening row over pay.
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The graduate job market has recovered to its pre-recession peak, with a 12% rise in the number of jobs for university leavers, a report says.