Simpson trained with West Brom earlier this month after failing to find a new club in the wake of his release by Leicester City, with The Sun reporting that a permanent contract could be offered by the Baggies.
But he has since moved on to Huddersfield, whose manager Danny Cowley is hoping to tie up a full-time deal for the 32-year-old.
Speaking to The Examiner on Thursday, Cowley said of Simpson, who attended Huddersfield's defeat against West Brom on Sunday: "He has been in the building now for about 10 days and we are really keen to be able to get something done.
"I know the chairman is working exceptionally hard to try and make it happen and, fingers crossed, we will be able to get it over the line."
Asked about Simpson's fitness, should he indeed agree terms with Huddersfield, the new Terriers boss added: "Well, I think he played around 10 to 12 games last year at Leicester in the Premier League - not so many at the back end of the season.
"But he played the Chelsea game, last game of the season, coming off the bench in that, which shows what the club and Brendan Rodgers thought of him.
"He has also trained at West Brom prior to coming in to us, so he has had about nearly four weeks now on the grass, so he is close to match fitness.
"Obviously he is just short of game time and that's something which we will hopefully, if we can make it happen, be able to provide for him."
Simpson won a Premier League title with Leicester and has on his CV three promotions to the Premier League, with Sunderland, Newcastle United and Queens Park Rangers.
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