Davenport, a 23-year-old midfielder who was released by Blackburn Rovers in the summer, is training with Hearts this week in an attempt to win a contract with the Tynecastle club.
“We brought him up to have a look at him,” Neilson said during his pre-match press conference ahead of the first Edinburgh derby of the season at Easter Road on Sunday.
“We are still assessing him at the moment and we will probably make a decision at the end of this week and over the weekend.”
Hearts are also reportedly keen to sign the South Korea international forward Seung-woo from Suwon FC, with talks said to be ongoing regarding a potential move for the 24-year-old former Barcelona player.
“We have been linked with loads of players recently,” Neilson said.
“I’m not going to comment on other teams’ players but we are looking in those areas to try and strengthen.”