The German MEP told this website the next President should come from either his group or the Socialists.
He said he supports a continuation of the agreement he signed with Martin Schulz, the current President.
This arrangement, which has existed for many years, has led to fresh accusations of a "stitch up" between the two big groups.
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But, speaking in Parliament on Wednesday, Weber said, "We should remember that the Liberals lost a lot of seats in the last European elections and are not the force or as powerful as they once were."
His comments will be seen as a setback to any ambitions Verhofstadt has to take over Parliament's most senior post from 1 January.
Weber said that, according to German press reports, Schulz is being lined up to become that country's next foreign Minister.
If so, that would pave the way for an EPP group candidate to take over the presidency - as Weber said had been originally agreed - for the second half of the current legislature.
Weber told a press briefing that the EPP group will make a final decision on its candidate or candidates at a meeting in Strasbourg on 13 December.
"I believe that, as the two biggest and most important groups, the EPP and Socialists have a specific responsibility here. We have invested a lot in this agreement and it should continue.
"I also believe that, under the terms of the agreement I signed with Schulz, we have the right to the presidency from 1 January."
He refused to divulge if he would be a candidate for the post.
Meanwhile, the group re-elected Weber as its leader until 2019.
Weber, who has chaired the group since 2014, was re-elected by an overwhelming majority, with 97.8 per cent of the votes.
Weber, 44, has been a MEP since 2004 and held various positions since then, including group coordinator in the civil liberties committee, and Vice-Chair responsible for the working group on legal and home affairs.
He succeeded French farmer Joseph Daul in 2014 as leader and is also the deputy Chair of his party, the CSU.