The approach followed talks in the senior ranks of Labour about a potential candidate to replace Goff, who has been Auckland Mayor since 2016.
Goff, a former trade minister, is tipped to be appointed as the Ambassador to Washington at the end of his term as the Government tries to boost its trade relationship under President Joe Biden.
The Auckland mayoralty is an important post for Labour as the Government undertakes significant reforms.
Those include a review of local government and the Three Waters amalgamation plan to vest water infrastructure and assets owned by councils in four regional entities.
Goff is opposed to signing Auckland Council up for those water reforms, which would see the Auckland and Northland regions water governed by one body.
The centre-left has held the mayoralty since 2010 under Len Brown and then Goff, who stood as an independent. Shearer has similar centrist political views to Goff on many issues.
The centre-right has struggled to find a strong candidate for the mayoralty in recent elections.
This time round, National MP Mark Mitchell has confirmed he is considering have a run.
The Whangaparaoa MP has still not ruled it out, saying on Friday he was yet to make any decisions.
His former colleague Paula Bennett has said she had absolutely no intention of standing, despite repeated speculation about it.