The centre-back has been on loan at the Stadium of Light in the second half of the season and is keen to earn a full-time switch to the Black Cats over the summer.
Davenport has played just six times for Ricky Sbragia's side but has featured heavily over the last few weeks and it's thought they are ready to offer him a deal ahead of next season.
The 26-year-old has a year left to run on his Hammers contract and has attracted interest from Wolves and Birmingham but appears set on a move to Sunderland.
He told the Sunderland Gazette: "The main focus for me is to do the job for Sunderland, then we'll get to the summer and then see what the situation is.
"But I'd definitely sign if Sunderland wanted me in the summer. I've really enjoyed myself here and I settled in really quickly.
"Whatever happens, I won't be at West Ham next season. I've got another year left on my contract but I'll be moving on in the summer."