The company has approached a number of non-roster agencies with a request for information regarding the brief.
Pitches are due to be held later this year, and the successful agency is expected to win a place on J&J's advertising roster.
Agencies already on the roster include Abbott Mead Vickers BBDO, DLKW Lowe and JWT London.
Brands in J&J's Suncare range include Piz Buin and Johnson's Facial Sun Protection.
There is not currently an incumbent agency on the Suncare account, instead J&J has used agencies from its existing advertising roster to work on campaigns on a project basis.
The last major campaign for Piz Buin was launched in 2007, created by Lowe London and used the line "Hours of gorgeous protection".
Piz Buin recently launched a Mountain range, which has been specifically designed to give sun protection in high altitude conditions.