There are a number of possible prepositions, but my instinct says 'into'.

Hello Sasha! You could say: Before signing the contract, the client wished to make some enquiries "about" the matter. or Before signing the contract, the client wished to make some enquiries "regarding" the matter. or Before signing the contract, the client wished to make some enquiries "around" the matter. or Before signing the contract, the client wished to make some enquiries "about the facts pertaining to" the matter. (This is more formal and might be better when speaking about signing a legal document). I hope that helps!

Thank you very much, Stephen! I've just added the screenshot https://www.dropbox.com/s/6ny3jut174nwnz4/Screenshot_29.png?dl=0

Hi Sasha, you could say: "...the client wished to make some enquiries ON the matter." You could also say: "...the client wished to make some enquiries REGARDING the matter." There is a third option too: "...the client wished to make some enquiries IN RESPECT OF the matter."