Boro confirmed they had accepted an offer for Tavernier from a top-flight club, believed to be Bournemouth.
“We have to get on with it, the club did everything to try and keep Marcus,” said Wilder.
“When you get down to the nitty-gritty of the move and him wanting to move, I think it just became a case of negotiating the transfer fee, which is above me.
“But as I say the club tried everything they could to keep Tav, and now we have to look forward.”
Tavernier is a product of the club’s Academy and made 155 appearances after making his debut against Scunthorpe United in the EFL Cup in August 2017. He scored the first of his 18 goals against Bournemouth a few weeks later in the same competition.
Tavernier’s aniticipated exit will be offset in some way by the previous day’s arrival of Brentford striker Marcis Forss on a four-year deal and, when asked about how he felt the club’s summer transfer activity had gone, Wilder replied: “It’s not been an easy window, as everybody knows.
“But the Premier League clubs are not done and that gives us a little bit of confidence and comfort knowing that we will get done what we need to before the window ends.
“We’d have liked to have got more things done by now, but for numerous reasons we’re not set, but I don’t think anybody is set.
“Whether that is over-priced players, or a lack of the kind of players that we want – there’s no point just stockpiling players, I’m not about that – we’ll just have to keep working away.”
On today’s visitors and their manager Steve Bruce, Wilder was quick to label West Brom among the promotion favourites.
“What Steve has achieved as a player and as a manager is fantastic – I wish I’d had his career as a manager,” added Wilder.
“You’ll have to go along way inside the game of football to find a more respected guy than Steve.
“I don’t think he’s gone to West Brom not to get out of this division and rightly so because we’re talking about parachute payments and the size of that club.
“And West Brom should be pushing at the top to get out of this division, with the players they have managed to bring in and those they have retained.
“I’m delighted it’s a really tough first game.”
Boro will be without Paddy McNair as the defender misses out through a suspension carried over from last season when he was dismissed on the final day against Preston. There are no other fresh injury concerns for Wilder, with new signing Forss possibly featuring at some point.
Bruce could hand out debuts to several new signings, with Jed Wallace set to make his Baggies debut after joining the club from Millwall over the summer.
John Swift and Okay Yokuslu are the other new arrivals who could also be in contention, although Kean Bryan is still a long-term injury concern and will be sidelined.