The east London club are also close to finalising a deal with Israeli defender Tal Ben-Haim.
Avram Grant's squad are currently in Germany for a training camp, which saw them beat Borussia Monchengladbach 2-0 in Ruhpolding on Saturday.
The new Hammers boss has been quick to strengthen a group which only narrowly escaped relegation last season, bringing in Frederic Piquionne, who he had worked with at former club Portsmouth, and Mexico winger Pablo Barrera as well as ex-Aston Villa midfielder Thomas Hitzlsperger.
However, long-term target Dorrans will remain at West Brom after the Scotland midfielder agreed a new four-year deal, while Cole, who came through the youth ranks at Upton Park, has moved on a Bosman free transfer to Liverpool, despite the Hammers having offered lucrative terms to their former trainee.
Nevertheless, Grant should still land at least one more player before the new Premier League campaign begins at Villa Park in just over three weeks' time.
Ben-Haim looks set to be reunited with Grant, his manager at both Chelsea and Fratton Park, with a compromise over salary reported to now have been reached.
A deal for Remy, however, may prove more testing to conclude.
Liverpool remain very much in contention as Nice hold out for a substantial cash offer with the Hammers looking to pay off the £12million transfer fee in instalments.
"For the moment it's the same status quo, but if Liverpool make an offer things will be done swiftly," Nice president Gilbert Stellardo told L'Equipe.
"The amount (offered by West Ham) suits me, but not the payment method.
"West Ham want to buy Loic because they think he could rapidly be worth 10million euros more if he plays in England.
"They don't want us to pay the full sum all at once."
The Hammers squad - which now also includes those who were away at the World Cup, such as England duo Robert Green and Matthew Upson - will play Greek side Panathinaikos before they return to England and head to MK Dons on July 28.
A trip to npower Championship side Ipswich and the visit of Spanish outfit Deportivo la Coruna completes the pre-season schedule.
Radoslav Kovac has been linked with a move to Olympiacos, but for now, the Czech midfielder is fully focused on getting off to a flying start with the Hammers.
"The new manager has been good with us and we all want to do our best for the club," he said on www.whufc.com.
"We have some new players like Thomas [Hitzlsperger] and we will see where we are with the Premier League. It is important to start well.
"I am happy to be back with everyone again. It was very good to have the time off, we needed it after a tough season but I want to get going again."