Alexander still played in Sunday's game against the Philadelphia Eagles because the suspension isn't expected to be handed down until later in the week, the source confirmed to Schefter.
Alexander addressed the situation after the Buccaneers' 45-17 win.
"I'm just waiting on the appeal process, but it's my fault. I didn't know," he said. "I have never been in any type of trouble or anything, so it was a surprise to me. I didn't know what I took or anything. It's in God's hands, so I'm just waiting for the appeal process, and whenever that is, and if I happen to get suspended for a game, then I'm going to come back even stronger."
The Tampa Bay Tribune was first to report Alexander's impending suspension. The rookie linebacker will have the chance to appeal the suspension.
Alexander rose to national prominence earlier this month when he recorded 11 tackles, an interception and a fumble recovery to lead the Bucs to an upset victory over the Atlanta Falcons less than two days after his 17-year-old brother was killed.
Alexander was given the game ball by his teammates following the 23-20 overtime victory and announced three days later that he was dedicating the rest of the season to his brother.
A fourth-round draft selection, Alexander is Tampa Bay's second-leading tackler with 63. He also has two sacks and two interceptions this season.