Hammers manager David Moyes had two bids, the last amounting to about £8million, for the Bournemouth star rejected during the January transfer window.
But Arter, 28, remains a major target for Moyes, who intends a significant shake up of his West Ham squad this summer should he succeed in keeping the club in the Premier League.
The Scot immediately identified midfield as an area which needed major improvement when he took over at the London Stadium in November – and the strong-running Arter is a player Moyes still believes he can prise away from the South Coast.
Moyes did sign Portuguese midfielder Joao Mario on loan from Inter Milan in January but the Italians want £38m to make the move permanent in the summer which is way out of West Ham’s price range.
West Ham face Swansea at the Liberty Stadium on Saturday, lying just three points above the Premier League relegation zone and having won only one of their last five games.