West Ham have made a £13m offer for Chelsea defender Emerson and proposed a £10m bid for Bruges captain Hans Vanaken.
Emerson has offers from Nottingham Forest and Atalanta while Juventus have been monitoring developments, highlighting that the 28-year-old is not short of options.
Meanwhile, manager David Moyes is hopeful the club can conclude a deal for Paris St Germain centre-back Thilo Kehrer this week if they can overcome some financial details.
However, a deal for goalscoring midfielder Vanaken will prove difficult with Bruges wanting to keep their captain.
Moyes is a long-term admirer of the 6ft 5ins Vanaken and has tried to sign him before, however, the Belgian side say the 29-year-old is not for sale.
Manchester United defender Eric Bailly is another player that remains on West Ham's radar though they face competition from Marseille on that front.
Bailly has not featured in either of Erik ten Hag's squads in the opening two games of the season at United, and now seems to be some way down the pecking order of central defenders.
Moyes is keen to bring in defensive reinforcements after being forced to play Ben Johnson as a makeshift centre back in West Ham's first two matches due to injuries.
The Irons did sign Nayef Aguerd from Rennes in June, but he picked up an ankle problem in pre-season which required surgery, and has not played since.
Moyes admitted his West Ham team looked 'stale' when they were beaten by Manchester City on the opening day, and he does not appear to have changed his mind after they lost again to Nottingham Forest on Sunday.