"Things have been really busy," Gillis told the Canucks website Saturday. "Getting a lot of calls, and a lot of discussions, and we'll see where it turns out."
The Canucks settled on Schneider as their No. 1 goalie heading into the 2012-13 season and have been looking to trade Roberto Luongo.
Gillis said when a team called to ask about Schneider, "You have to listen."
"I was a little bit surprised," Gillis said of the discussion becoming public. "But sometimes that happens; a team calls you about a player and then suddenly it gets out. We just have to live with it and keep moving ahead."
Sportsnet reporter John Shannon tweeted there is "plenty of interest" in Schneider, who has two years at $4 million each remaining on his contract before he would become an unrestricted free agent.
"We're listening to proposals; you never say never about anything," Gillis said. "We'll have to wait and see what happens."
Schneider was 17-9-4 with a 2.11 goals-against average and a .927 save percentage during the regular season. The 27-year-old was injured at the start of the Stanley Cup Playoffs but returned to play in the final two games of a four-game sweep by the San Jose Sharks.
"I think he's a very good young player," Gillis said of Schneider. "And teams are after good young players all the time. So, I'm not surprised; we get calls all the time about our young players and he's one of them, so it's not surprising at all."
Luongo, 34, has nine years remaining with an annual salary-cap charge of $5.33 million. At the NHL Trade Deadline, he admitted he would be difficult to trade because of that contract. He was 9-6-3 with a 2.56 GAA and .907 save percentage in 2012-13.
Asked Saturday if he would use a compliance buyout on Luongo, Gillis replied, "I just don't know yet." Doing so would cost two-thirds of the remaining value of the contract paid over twice the term (approximately $32 million over 18 years), but it would remove the annual charge from the Canucks' salary cap.
Gillis would not say specifically what it would take for him to trade Schneider, or if there was any interest in Luongo.
"You have to listen to what the proposals may be and act accordingly, and that's what we're doing," Gillis said.