Hill has joined Scottish Premiership-bound Rangers on a free transfer after leaving Queens Park Rangers at the end of the 2015-16 season.
The 37-year-old defender has a lot of experience, but he is still looking to learn more from Everton assistant manager Weir.
Like Hill, Weir also joined Rangers in his mid-30s, but he went on to establish himself as a club legend.
The former Stoke City and Crystal Palace defender has shared his admiration of Weir, and has outlined how he plans to make the most of the opportunity to work with him.
"I've got massive respect for Davie because I know how hard it is to stay in this game," Hill told The Brechin Advertiser.
"It can eat you alive at certain points. I think he stayed in until he was 41 and played in a UEFA Cup final. That's an unbelievable achievement.
"I'll be pestering him on a daily basis to get some inspiration from him. I'll also be talking to him about what they need from me to be a Rangers player.”
Rangers won the Scottish Championship title with ease last season and got automatic promotion to the Scottish Premiership for the 2016-17 campaign.
The Gers have been quite busy in the summer transfer window, signing Hill and former QPR midfielder Joey Barton among others.
Celtic remain the strongest team in Scotland, though, and will start the 2016-17 Scottish Premiership season as favourites to win the league title.