Suter's agent, Neil Sheehy, said Sunday those suitors shouldn't expect a decision from his client anytime soon.
"I am fielding many calls and methodically returning each one by one. It is going to take some time," Sheehy told ESPN.com's Pierre LeBrun.
Detroit, Philadelphia, Pittsburgh, Minnesota and Carolina are among the teams making a push for Suter.
Parise, meanwhile, said he thinks he'll make a decision by Monday after opting not to sign a deal to stay with New Jersey and becoming a free agent.
"I'm not at liberty to say right now who's in or anything like that," Parise told reporters. "But just because it got to noon, by no means does that say that the Devils aren't one of my top teams."
Parise's agents, Don Meehan and Wade Arnott, didn't allow teams to have face-to-face meetings with the star forward despite at least one being in the area that was hoping to have an audience with him.
"He has narrowed the list of teams to a small select group," Meehan wrote in an email to The AP. "The Devils are still in consideration."
Detroit coach Mike Babcock traveled to Toronto to be part of the franchise's planned pitch to Parise.
"We're going to get in the mix with him and see what we can do," Babcock told The AP in a telephone interview minutes after teams could begin talks with free agents. "We're going to do everything we can."
The Red Wings, though, had to cancel their plans to send a contingent to talk with Parise in person.
"I called Don Meehan at 12:01 and heard back from him at about 12:30 and he said based upon the phone calls, it would almost be physically impossible to meet with all the interested teams," Detroit general manager Ken Holland said in a telephone interview. "We were prepared to meet with them, but we can to cancel our plans."
Holland confirmed he made offers to both Suter and Parise, the best defenseman and forward on the market.
Detroit, though, isn't the only NHL team with a lot of money to spend in free agency.
The Minnesota Wild posted a message on the team's Twitter account, announcing they submitted contract offers to Parise and Suter, within the first hour of free agency.
Carolina Hurricanes general manager Jim Rutherford confirmed he called agents for Parise and Suter to offer deals.
"We're just playing a waiting game," Rutherford said.
He's not alone.
Teams will have to wait before simply finding out if they can have a face-to-face meeting with Suter.
Sheehy fielded "countless phone calls, emails and messages" from teams hoping to sign his coveted client. Sheey planned to respond to each team to gather more information before presenting options to Suter and deciding whether to meet with interested clubs.
New Jersey attempted to re-sign Parise and Nashville tried to keep Suter before they hit the market, but the Devils and Predators ended up being relegated to competing with other offers. Parise and Suter are both 27, potentially entering the prime of their careers.
"Sutes is in an enviable position that he has got everybody coming after him and the money obviously is going to be whatever he wants it to be," Predators general manager David Poile told reporters Sunday. "I'm hopeful that he is going make his decision based on his team and his teammates and the city of Nashville compared to everything else that is out there."
Information from The Associated Press was used in this report.