The Daily Star says fed-up Blues stars have lost faith in Mancini and want Vieira to replace him.
It is understood Vieira is the man the players want to take charge if the Italian leaves. Vieira, 36, is a popular figure among the first-team squad.
The former World Cup winner has been working as the club's football development executive.
He started doing his coaching badges in the summer and completed his 'B' licence. He is working towards his 'A' and 'Pro' licences.
His lack of coaching qualifications would restrict him to a 'caretaker role' initially but under Premier League rules Vieira would be allowed to manage for 12 weeks.
Blues stars have made him their 'unanimous choice' to take over on an interim basis should the club's Abu Dhabi owners decide to make a change.