The Hammers tied up their first signing of the summer last week when they landed right back Ryan Fredericks from Fulham, with the defender signing on a free transfer.
New manager Manuel Pellegrini will be expecting plenty more business to be done though, and signing a new goalkeeper has been a priority for the Hammers ahead of next season.
Joe Hart has returned to Manchester City after a disappointing loan spell, whilst Adrian's future remains up in the air amid rumours of a potential return to Spain.
Now, The Mirror report that West Ham could be nearing their second summer signing, as a deal for Swansea City shot-stopper Lukasz Fabianski is progressing well.
Whilst Swansea want a fee of around £9million for Fabianski, West Ham hope to strike a deal at £7million, handing the former Arsenal flop a return to London.
Fabianski, 33, has starred for Swansea since his 2014 move from Arsenal, and was always likely to head back to the Premier League following the Swans' relegation at the end of the season.
As a proven Premier League goalkeeper, Fabianski should be a safe pair of hands for West Ham, who hope to get a deal done before Fabianski's World Cup campaign begins with Poland next Tuesday.