Radwanska came into the Australian Open with titles this season at Auckland and Sydney, where she beat Li in the semifinals, but continued her poor conversion rate in Grand Slam quarterfinals. She's advanced further only once in seven tries at Wimbledon last year, when she lost the final to Serena Williams.
Sixth-seeded Li will play the winner of Tuesday's later all-Russian quarterfinal between No. 2-ranked Maria Sharapova, who conceded only five games in her first four matches at Melbourne Park, and Ekaterina Makarova.
The quarterfinals on the other half of the draw will feature American teenager Sloane Stephens against Serena Williams, who is aiming for a third consecutive major title, and defending champion Victoria Azarenka against two-time major winner Svetlana Kuznetsova.
Stephens has heard a lot of advice from Serena Williams. Pointers on her groundstrokes, and even on her grunts.
It's been mostly gentle encouragement, occasionally spiced with headline-making comments from Williams, who has predicted the 19-year-old American will one day top the women's rankings.
As Stephens learned earlier this month, though, it's one thing to play with Williams, another to play against the 15-time Grand Slam champion.
With a comeback 6-1, 3-6, 7-5 win over Bojana Jovanovski on Monday, Stephens qualified for her first quarterfinal at a major tournament. Williams has already played at that level 34 times.
"It will be tough, obviously. It's quarters of a Grand Slam," Stephens said. "There won't be that, like, first time, 'Oh, my God, I'm playing Serena.' That's kind of out of the window now. So that's good."
Williams and defending champion Victoria Azarenka advanced Monday, losing just four games between them against Russian rivals. Williams beat No. 14 Maria Kirilenko 6-2, 6-0, and Azarenka defeated Elena Vesnina 6-1, 6-1.
On the men's side, No. 2 Roger Federer and U.S. Open champion Andy Murray stayed on course for a semifinal in their half of the draw.
Federer won 6-4, 7-6 (4), 6-2 over big-serving Canadian Milos Raonic, advancing to the quarterfinals at a Grand Slam for the 35th consecutive time, while Murray took advantage of Gilles Simon's fatigue for a 6-3, 6-1, 6-3 victory.
P Tuesday, noon (tape), 7 p.m.
TV • ESPN2