The Baggies are turning their attention to their strike-force for the final two weeks of the transfer window as Slaven Bilic targets more signings.
And Albion were keen on Ajeti, a 22-year-old who scored 21 goals in 43 games for the Swiss outfit last season.
But they were unwilling to meet the wage demands of the Switzerland international, which the board felt were unreasonably high.
The Baggies retain an interest in Ajeti, but are looking elsewhere for now.
Albion completed their fifth deal of the summer over the weekend, bringing midfielder Romaine Sawyers back to the club in a deal worth £2.9million.
But Bilic and the board are keen to bolster their attacking options before the window shuts next week, and still have money to spend from the sale of Salomon Rondon to China.
They have been looking at wingers and strikers to help ease the burden in front of goal currently on Kenneth Zohore, Hal Robson-Kanu and Oliver Burke.
“It’s pure mathematics from last year, you know better than me how many players we lost,” said Bilic, before Sawyers’ arrival was confirmed.
“We’ve brought in maybe half of the number. I’m not saying we lost nine, we’re going to get nine, but we can’t lose nine and get four.
“Especially when seven of those nine were regular big-time players. We need a few more, not only the number but the quality.
"We’re trying to, it’s not very easy to get the players. But we’ve got some good ones so far, I’m really happy.
"Hopefully we’re going to get a few more, but not crazy numbers. Only a few, because we need a squad.”
If there are any funds left over from their attacking additions, the Baggies may look to add one more player in defence with Ahmed Hegazi missing the first six league games of the season.
But the priority is up front, where the Baggies have so far only signed Zohore in a deal worth up to £8m to replace the loss of Jay Rodriguez and Dwight Gayle, who scored 45 goals between them last season.
Bilic was concerned by the lack of goals in Albion’s final warm-up game of pre-season, a goalless draw with Bournemouth, but was pleased with the build-up play behind the strikers.
“I’m long enough in football, I’ve seen teams being fit and great one week before the start and then on matchday, no good, and the other way round,” he said. “But I can only comment on the match and can only say very positive things.
“What pleases me is when you want to give a chance to the other guys from the bench, if you change three or four or more, normally the standard and intensity and the rhythm and the quality drops. It didn’t drop.
“That’s the thing that pleases me the most. They even raised the level.”