The newly-appointed head coach has been linked with the Albion defender he picked for his senior national squad when he was Italy boss.
However indications from the city’s leading newspaper and website are that, while certainly interested in Coppola, Gattuso has preferred targets for the challenge currently facing his new club.
The suggestion is that Coppola would be a player he would look at again in the future.
Lazio missed out on Europe as they finished ninth last season and are selling centre-back Alessio Romagnoli to Qatar while looking to reduce the age profile of their squad.
For now, Lazio are said to be focussing more closely on Arnau Martinez at Girona and Sergi Domiguez at Dinamo Zagreb.
They are also believed to be weighing up a loan offer for Italian youngster Fabio Chiarodia of Borussia Monchengladbach.
Some context as to the financial power of Serie A is perhaps offered by the fact a release clause of about £6 million for Martinez is said to represent “a bottleneck” for Lazio.
Gattuso has been on holiday in Marbella but has been keeping abreast of his transfer options.
Albion’s own position in the centre of defence is pretty fluid.
Jan Paul van Hecke and Adam Webster have left and the club are keen on Luka Vuskovic at Spurs and Michael Svoboda of Venezia.
Igor Julio could also depart but he was not a member off matchday squads last season after returning from loan at West Ham.