Weems, an all-Euroleague first-team selection in 2013-14, gets a two-year, $5.8 million deal with the second year a team option. He will make just over $2.8 million this coming season.
Price agreed to a one-year deal at the veteran's minimum of $1.5 million.
The person who confirmed the deals sought anonymity because free agent contracts cannot become official until Thursday.
The 6-foot-6 Weems, who turned 29 on Wednesday, has played the last four seasons in Europe, the last three in Russia.
Last season, he averaged 13.1 points in Euroleague play and 11.5 points in the Russian league, shooting a combined 40 percent from 3-point range.
Weems, a second-round pick by Chicago out of Arkansas in 2008, played in the NBA from 2008 through 2010. He played 12 games for Denver in 2008, 69 games for Toronto in 2009 and 59 for the Raptors in 2010. He averaged 9.2 points in his last season in Toronto.
Weems is one of three shooting guards on the Suns roster and could get the most playing time since the other two are youngsters Archie Goodwin and rookie Devin Booker.
Price has played for six teams in 10 NBA seasons. He will be a third point guard who will play when both Bledsoe and Knight are off the court, a situation the Suns hope rarely occurs.