The company is seeking one agency for creative strategy for eight of its core brands: Durex, Nurofen, Vanish, Finish and Gaviscon across Europe; and the "local hero" brands Lemsip, E45 and Optrex in the UK.
Separately, Reckitt Benckiser will also appoint an agency to work on digital production for all of its 19 "powerbrands" across Europe.
The procurement-led pitch is thought to have no intermediary and the successful agencies will be paid about £1.2 million in fees between them. They will be appointed to start work in May.
The brief is thought to involve creating digital advertising and building websites.
TMW, which has handled digital advertising for Reckitt Benckiser’s UK brand portfolio since April 2013, is involved in the process.
A spokeswoman for Reckitt Benckiser said it "does not comment on speculation about pitches".
In March, TMW created an online film for Durex that showed couples ignoring each other while absorbed in gadgets. The spot tied in with Earth Hour, the movement that encourages people to stop using electricity for one hour. Durex invited people to "turn off" to "turn on" and enjoy their partner.
Reckitt Benckiser is also currently reviewing the advertising for three of its biggest brands: Finish, Air Wick and Clearasil, held by Havas Worldwide, its global creative agency.
Reckitt Benckiser was the 13th-biggest-spending advertiser in the UK in 2013. It spent nearly £70 million, according to Nielsen.