Delph came through the ranks at Leeds and rejoined the Championship club on an initial month-long loan deal on Friday.
Grayson wants the midfielder to remain part of the Whites' bid for promotion to the Premier League and is intending to hold further talks with Villa.
"We tried to do it to the end of the season and Fabian wants to come till the end of the season but Villa have only agreed to a 28-day loan," said Grayson in the Yorkshire Evening Post.
"Over the next 10 days we'll be in negotiations, trying to extend that to the end of the season. That would give us stability and the player stability. But I have to be happy to have him for at least seven games with the potential for him to be here a lot longer.
"In the 10 days before the window shuts we can turn it into a standard loan which is what we want to do. If not, his 28 days can go on to 93 days but Villa would be able to recall him.
"There's still a bit of negotiating needed to sort that out."
Meanwhile, Grayson also rejected rumours that Delph has been signed to replace Jonny Howson, who is set to join Premier League side Norwich City.
He added: "This isn't a deal that's been done in the last 24 hours. We'd spoken about it before Christmas and we were at the mercy of Villa.
"It was only in the last two or three days that they decided what they wanted to do with him.
"It's a signing we were looking to do regardless of Jonny leaving. This is an expensive deal for Leeds United but it shows our intent."