The Celtic League clash comes eight days ahead of Scotland's Calcutta Cup clash against England at Twickenham, their opener in this season's Six Nations Championship.
Cusiter has been sidelined since dislocating a shoulder in Scotland's match against the Pacific Islands during November.
However Hadden has until Tuesday to select his team and will pay close attention to how Cusiter fares on Borders' trip to Wales, if selected.
The scrum-half position is Cusiter's for the taking if he can demonstrate a good level of fitness, given that injured Mike Blair will certainly miss the England match.
He impressed in training with Scotland over the past two days, and Hadden said: "We're delighted with the progress that Simon Webster, Chris Cusiter and Alli Hogg have made, and the amount of involvement they have had this week.
"But we still have to cross some t's and dot some i's before they become available for selection."
Cusiter, 24, told The Herald: "With Mike's situation, I knew there were places up for grabs and I hope I can get a game in this week and make myself available.
"I'll be lacking match practice, which is fairly obvious, but guys like Simon (Taylor) have come straight back into the (Scotland) side after a long injury and they've come through it."
Allister Hogg could also line up for Edinburgh against Connacht on Friday, however he is not expected to feature at Twickenham.