Man United and Chelsea are both reportedly weighing up bids for Gregory van der Wiel after the right-back confirmed he will be sold by PSG.
Wolves have completed the signing of Adam Le Fondre on a season-long loan from Cardiff.
The 48-year-old is overseeing first-team affairs following Albion's decision to sack head coach Steve Clarke and his assistant Kevin Keen.
In their statement announcing the news on Saturday, the Baggies gave no indication of how long they might take to appoint Clarke's successor, saying simply that joint-assistant head coach Downing would assume control "whilst the club considers all available options".
And Press Association Sport understands the anticipation is that Downing will still be in that position come the visit of Hull to The Hawthorns this weekend.
In that match, West Brom will be looking to halt a losing streak that extended to four games with the 1-0 defeat at Cardiff on Saturday which preceded the confirmation of Clarke's dismissal.
Albion are 16th in the table, two points clear of the relegation zone, and have won only seven of their 34 top-flight games in 2013.
The club are not making any comment regarding who their new permanent boss might be, but various people have been linked with the vacancy since Clarke's departure.
Prominent among them are Roberto Di Matteo - who guided West Brom into the top flight in 2010, was sacked by them in February 2011 and then went on to win the FA Cup and Champions League during a stint in charge at Chelsea in 2012 - and ex-Albion player Martin Jol, recently dismissed from his post as manager of Fulham.
Others being mentioned as possible candidates are Molde boss Ole Gunnar Solskjaer, ex-Schalke coach Ralf Rangnick, Real Madrid assistant coach Paul Clement, Mike Phelan - most recently Sir Alex Ferguson's assistant at Manchester United - and ex-Blackburn boss Michael Appleton, who has managed Albion on a caretaker basis before.
Both of the latter pair are also former Baggies players.
Clarke was appointed as West Brom's head coach in June 2012 - taking on his first managerial role - and went on to guide them to their highest-ever Premier League finish of eighth last season.
Albion lost the services of their 2012-13 top-scorer over the summer when Romelu Lukaku's loan spell with them ended, and the players subsequently added to the Baggies attack have not really truly sparkled.
Nonetheless, the club managed to achieve some notable results this term, such as a 2-1 win at Manchester United and 2-2 draw at Chelsea.
Giving his reaction to Clarke's dismissal, former West Brom player Cyrille Regis, quoted by the Birmingham Mail, said: "I am shocked - I didn't see that coming at all.
"If you're looking at the bigger picture then I thought he might get longer but it shows the pressures of staying in the Premier League.
"Certain players haven't come to the level that was expected either, which didn't help him.
"Martin Jol would be a strong candidate. He knows the club, he knows the continental system Albion have in place and he'll know the parameters Albion work under because he had them at Fulham.
"He's managed at the top level across Europe so he ticks many boxes."
Regis, who played for Albion at the same time as Jol, added: "I think with the position Albion are in they need a 'been there, done that' manager.
"It needs someone with experience to stop the slide."
Meanwhile, it is understood that West Brom will be dealing with Saido Berahino internally, with it looking likely that he will be disciplined, after a foul-mouthed message was posted on the Albion striker's official Twitter account.
The message, which appeared on Saturday after news of Clarke's sacking broke, said: "Wow i cant believe it, that's f****** s***!!!!"
A subsequent message appeared saying: "By the way i apologise for my last tweet. This was related to a personal issue and nothing else. Please don't take the wrong way."
Another then said "I just had a flat tyre", accompanied by a picture of one. Both that message and the first one have been deleted.