Following discussions with Public Health England (PHE) and Orient, the EFL was today set to confirm that the third-round tie in east London would not go ahead as planned, with PHE due to make the final decision.
Under the competition's rules, Orient will forfeit the tie, with Spurs progressing to next week's fourth-round against rivals Chelsea or Barnsley.
However, Orient chairman Nigel Travis was this morning in discussions with EFL chief executive Rick Parry in an attempt to have the fixture rescheduled.
"We have talked to the EFL and we are assuming this is a postponement," Travis told talkSPORT.
"We’re going to lose £2.5million this year, also anything that can help us reduce that loss is clearly important. We are struggling, so games like this have to take place."
Under EFL rules, clubs receive a bye if their opponents cannot fulfil a fixture, while Spurs are currently in the midst of a run of nine games in 22 days.
But Travis continued: "We have a great relationship with Spurs, we’ve looked at their fixtures and this game can be played. There has to be a will, because this game is worth £150,000 profit to us, so we have to play this game.
"I recognise the guidelines and we have spoken to the league about extenuating circumstances. I can assure you, we will not be the first club facing this this year. Even Man City had a player test positive recently.
"We expect this game to be played and we will campaign very strongly for this game to be played. Clubs of our size depend on games like this."
Tottenham paid for Orient's players and staff to be tested on Saturday after their 2-2 draw with Mansfield because there is no requirement for regular testing of EFL clubs. Other top-flight clubs, including Chelsea and Liverpool, have paid for lower-league opponents to be tested but there is no obligation for EFL clubs to accept such offers.
The Breyer Group Stadium and Orient's training ground is set to remain closed while the EFL works with the club and relevant authorities to review its Covid-19 secure procedures, while a decision on their league match at Walsall, due to take place on Saturday, will be determined later this week.
Orient's recent opponents Plymouth Argyle and Oldham Athletic were all informed of the situation yesterday, with the EFL set to contact them to arrange testing, while Mansfield have lodged a formal complaint with the EFL and want to be reimbursed after spending £3,000 on testing kits for their players.
Tottenham lobbied the EFL to reschedule the fixture, believing it was unfair to have to play two midweek games, including Thursday's Europa League second qualifying round in Macedonia against Shkendija.
Jose Mourinho, the Spurs manager, has described the club's schedule as a "total joke" and "inhuman". He had been planning to start Dele Alli and Harry Kane tonight, the latter against the club where he made his professional debut and whose shirts he now sponsors.