Rice is set to undergo a medical with the Gunners after the Hammers accepted a £105million bid for the midfielder.
Leicester are expecting Barnes to follow James Maddison out of the club following their relegation from the Premier League.
Maddison signed for Tottenham in a £45m deal which was announced on Wednesday.
It is understood West Ham will also target two midfielders to replace Rice.
Joao Palhinha, Denis Zakaria, Amadou Onana and James Ward-Prowse are all thought to be on the Hammers's radar.
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Barnes, meanwhile, is viewed as one of the Foxes biggest assets and the Championship club could cash in as new boss Enzo Marseca shapes his side ahead of their upcoming campaign.
Both Newcastle and Aston Villa have shown an interest in the 25-year-old winger but West Ham look set to launch an offensive.
Barnes has made 187 appearances for Leicester, scoring 45 goals since making his debut in 2018.