Mr Story was refused an injunction which would have prevented the vacant seat being offered to Leeds councillor John Procter.
Leeds City Council chief executive Tom Riordan, acting as the European elections returning officer for Yorkshire, will invite Coun Procter to take the seat unless Mr Story mounts a further legal challenge on Thursday.
Mr Riordan said: “The judge dismissed Mr Story’s application and refused his request to appeal.
“However, I have agreed to grant him a further 24 hours to approach the Court of Appeal himself. Unless otherwise directed by the court, I intend to make the formal offer of the seat to John Procter at 4.30pm tomorrow.”
The High Court refused Mr Story’s request to stop Mr Proctor being invited to take the European Parliament seat left vacant by former MEP Timothy Kirkhope’s move to the Lords.
The row began after former MEP Timothy Kirkhope was awarded a peerage by outgoing prime minister David Cameron.
As the second candidate on the Conservatives’ Yorkshire list at the 2014 European elections, Mr Story was initially offered the vacant seat.
However, the Conservatives refused to endorse his candidacy and instead backed Coun Procter, who was third on their list in 2014.
European elections are conducted under proportional representation rules and on a regional basis.
Each party submits a list of candidates and, in broad terms, the more votes it receives the more of its candidates become MEPs.
At the last election in 2014, the Conservatives secured one of Yorkshire’s six seats which went to its top candidate, Mr, now Lord, Kirkhope.
If a Westminster MP steps down a by-election is triggered but in the case of an MEP resigning the seat normally goes to the next person on their candidates list.
In 2014, Mr Story was the Conservatives’ second-ranked candidate while Coun Procter was third.
The Conservatives have previously issued a statement describing Coun Procter as Yorkshire’s new MEP despite the fact he has not been offered the seat.
Mr Story told The Yorkshire Post last month he had told Mr Riordan he would be taking the seat, saying it was a “privilege to represent Yorkshire”.
He was the Conservatives’ candidate for Leeds North West at the 2015 General Election and in Wakefield in 2010.
Before entering politics, Mr Story represented Britain at the 1996 Olympics.