Initial reports following Wednesday's announcement suggested the club could face a bill of up to £24 million — which would have exceeded the £23 million paid to José Mourinho and his staff in 2007 — given Rosenior had signed a deal until 2032 worth around £4 million per season.
However, Sky Sports News and Fabrizio Romano have both confirmed that reports of a large pay-off are not accurate.
"There is a special clause in the agreement Rosenior made with Chelsea in January that will mean Chelsea avoid paying crazy money to the manager," Romano said.
"Chelsea are not going to pay the full salary for the long-term contract. There is a break clause in the contract. So financially for Chelsea, it's not going to be as bad as it would have been without this clause."
Sky Sports News added: "The settlement will be fair for him and fair for the club."
The news provides some financial relief for a club that disclosed a Premier League-record loss of £262.4 million last season. Missing the Champions League — Chelsea are currently seven points behind fifth-placed Liverpool with five games remaining — would represent an additional blow to the club's finances.
Sixth place could theoretically still be sufficient if Aston Villa win the Europa League while finishing fifth.