In a statement, the Lightning said they are aware of the allegations against Cole and are cooperating fully with the NHL on an investigation.
"Our organization takes these allegations very seriously," the team's statement read. "While we continue to gather more details, we have decided to suspend Ian Cole pending the results of an investigation. No members of the organization, including players, will comment further at this time."
Cole, in a statement released through his agent, Kevin Magnuson, said he "completely" denies the allegations and will cooperate with the league, the team and their legal departments in the forthcoming investigation.
"I take the allegations made against me today in an anonymous tweet very seriously,'' Cole said. "I look forward to clearing my name and demonstrating to the NHL and the Tampa Bay Lightning that these allegations are unfounded.''
The allegations against Cole were made in a statement posted to Twitter on Friday by an anonymous user. The person wrote that Cole groomed her for four years "starting when I was a minor in high school" and noted that Cole was "well aware of my age" at the time of the alleged sexual abuse.
The Twitter post did not specify where or when the alleged encounters occurred.
The Lightning would not clarify when contacted by ESPN whether the investigation into the allegations against Cole would be conducted by the team, the league or law enforcement.
A source told ESPN's Emily Kaplan that Cole's suspension is considered paid leave.
Cole, 33, is entering his 13th NHL season. He signed a one-year free-agent deal worth $3 million with the Lightning during the offseason and was expected to be in their lineup when Tampa Bay opens the regular season on Tuesday at the New York Rangers.
Cole, a Michigan native, made his NHL debut with St. Louis in 2010 and played five seasons for the Blues before being traded to the Pittsburgh Penguins. He also has played for the Columbus Blue Jackets, Colorado Avalanche, Minnesota Wild and Carolina Hurricanes.
The Associated Press contributed to this report.