The broadcaster's current £275m, four-year deal expires at the end of this season, and ITV is keen to secure reduced terms for the rights than it agreed in 2007.
According to The Guardian, ITV is close to agreeing a new two-year deal as there have been few other "credible" bidders in the running.
ESPN already has an existing agreement to show FA Cup games and some international matches, while the BBC recently cut back its sports rights budget by 15% in its Delivering Quality First cost-cutting initiative.
"As I understand it there is an air of resignation at the FA that they have to take it on the chin this time," said a source with knowledge of the talks.
"They have been preparing people and stakeholders [for a reduced deal] and the talk has been of at least a 20% reduction, with the range rumoured to run from 25% to 45% although the top end seems a bit extreme."
Sky, which held the FA Cup rights jointly with the BBC until 2007, is thought to have considered a new bid, but is not expected to come in with a late proposal.
However, there is a chance that Sky and ITV could share the rights in a similar way to which they currently do with UEFA Champions League games .
Channel 4 is not thought to be in the running, while Richard Desmond's Channel 5 appears to have also ruled itself out, despite recently losing the UEFA Europa League rights to ITV
The FA and ITV declined to comment on the report.