The Australian Open champion needs to win her quarterfinal match on Friday to take the top spot from Caroline Wozniacki.
"It didn't get into my mind," said Clijsters, who last held the No. 1 ranking in March 2006. "One more match, I guess. It will be tough. There's a couple of tough players who might get through."
Nadia Petrova of Russia, Andrea Petkovic of Germany, Yanina Wickmayer of Belgium and Sofia Arvidsson of Sweden advanced to the second round. Clijsters will play either Petrova or Jelena Dokic of Australia.
Clijsters got off to a sluggish start, dropping serve twice to give Barrois a 4-0 lead.
The top-seeded Belgian cut the deficit to 4-3, but Barrois converted her first set point when Clijsters sent a backhand volley into the net.
"She's a tricky player," Clijsters said. "She has a game that you don't see that often anymore on the women's tour. In the beginning, I just had to get a feel for it."
In the next set, Clijsters hit a cross-court backhand winner for a 3-1 lead. She double-faulted in the next game but broke Barrois with another backhand winner for a 4-2 lead, clinching the second set when a forehand from Barrois sailed long.
Clijsters was in control in the final set, although she had to save three break points in the third game.
"I wasn't worried after the first set," said Clijsters, who won the tournament in 2004. "I was sometimes overdoing a bit too much, I was trying to go for the winners a little bit too fast. But once I got that under control, I felt a lot more confident during the rallies."