The state-owned company, which employs 170,000 people has contacted regions around the country asking about interest in redundancy and part-time working.
The Daily Mirror has quoted an internal Royal Mail document which reads: "All teams have been given a 10% cash reduction target and so we need to focus not only on where we can deliver an excellent service with fewer roles but also opportunities to spend less cash."
Royal Mail, which has cut 50,000 jobs since 2002, is refusing to say how many redundancies it expects to make.
A Royal Mail spokesman said: "Royal Mail has made absolutely no announcement of job losses but we want to work with our people so that we take into account their wishes and preferences as Royal Mail continues to modernise and transform its operations to meet the challenges of falling mail volumes and intensifying competition, especially from email and the web as broadband use expands."
The Communication Workers Union, which has already said it will fight government plans to sell a minority stake in Royal Mail to a private company, attacked the measures.
Dave Ward, deputy leader of the CWU, said: "We are not opposed to modernisation but these are random savings which have not been properly worked out."