Connaughton on Monday underwent surgery on a fractured metacarpal sustained in the 131-107 loss to the Phoenix Suns on Thursday but should return before the end of the regular season.
The seventh-year shooting guard's injury leaves the Bucks short of backcourt depth, however, after they traded Donte DiVincenzo to the Sacramento Kings in a four-team deal that landed them Serge Ibaka.
Milwaukee, who are two games back of Eastern Conference leaders the Miami Heat, have consequently been heavily linked with a move for Dragic.
Dragic was traded to the Spurs by the Toronto Raptors last week but is set to be bought out of his contract by San Antonio.
ESPN has reported the Bucks will now be "aggressive in [their] pursuit" of the Slovenian guard as they look to secure a decent seeding ahead of the playoffs and their defence of the NBA championship.
The Bucks are 0-5 in exhibition games following a 107-97 defeat to the Brooklyn Nets on Wednesday, a clash in which Antetokounmpo had 24 points on the back of 6-for-21 shooting and grabbed 14 rebounds.
While such results may seem at odds with the Bucks' standing as one of the favourites for the NBA title this coming campaign, Antetokounmpo acknowledged there is plenty of time to find rhythm during the regular season.
However, the Greek superstar wants to see an improvement with some basics on the floor.
"It doesn't worry me," Antetokounmpo said. "What worries me is our habits and building good habits.
"Right now, we're not vocal enough. We're not urgent enough. We're not hungry enough. But at the end of the day, it's the preseason. We have the whole regular season to find ourselves."
Having featured for Greece at Eurobasket during the close season, Antetokounmpo has had his minutes carefully managed by the Bucks, with the Nets game just his second tune-up action.
While conceding to feeling "a little bit out of rhythm", Antetokounmpo is ready to once again set the tone for the Bucks.
"It's not going to be easy. I think sometimes your mind forgets and thinks it's going to be easy. But I feel like, me personally, you've got to come out and set the tone, play hard," he added.
"Build good habits, play to improve, play for the team. And as a team, we have to be hungry, we've got to play together, we have to defend better."
Grayson Allen (illness), Pat Connaughton (calf) and Wesley Matthews (ankle) were all absent against the Nets - the latter having sat out all of the Bucks' warm-up matches.
Khris Middleton will definitely miss their season opener against the Philadelphia 76ers having only recently returned to practice following wrist surgery.
But Antetokounmpo has confidence in the depth of the roster.
"Guys are down, who knows when they're going to get back, so this is who we have right now," he said.
"We still have a great team, but we're seven or eight days from the next game and when it really counts. We've got to buckle down and start trying to build good habits."
