Albion have so far signed Pablo Ibanez and Steven Reid - both on free transfers - plus Gabriel Tamas in a £800,000 deal.
Ashworth revealed the Baggies have several more targets on the horizon but will not sign players of insufficient quality.
He said: "We've got feelers and enquiries out on several different targets. There are conversations ongoing.
"If one of our top targets comes up and the deal is do-able, then we will do it.
"But we won't jump in for number six on our list for the sake of bringing in a player. We would be wise in holding on to get the right player."
Ashworth also admitted West Brom will look to make use of loan deals to help strengthen the squad which gained promotion from the Championship.
The Baggies have been linked with signing Liverpool striker David NGog on a temporary basis and Ashworth added: "The loan market is something we want to exploit.
"You maybe fill your squad and, once you have your main targets, you get a loan player in to fill a key position, perhaps.
"The chairman has alluded to it already, that for a loan signing, a Bosman, or a relatively low transfer fee, we can then bring in a player on higher wages."
Meanwhile, West Brom confirmed on their website that they are in talks with Villarreal over the possible transfer of midfielder Borja Valero away from The Hawthorns.
Valero has been given permission to train with the Spanish club and undergo a medical while discussions continue.