Jose Enrique has been signed to take over on the left and Glen Johnson has renewed his contract to stay at the club for two more seasons. Ryan Shawcross, Scott Dann, Gary Cahill and Per Mertesacker have all been mentioned as potential signings.
Now, a new target has emerged, as Dalglish's search has led him to Stamford Bridge.
The 29-year-old Brazilian joined Chelsea in 2007, making 84 appearances in his four seasons at the club. He fell out of favour somewhat when the Blues signed countryman David Luiz in January last season.
He is reported to be looking for a new club, and Liverpool is a very realistic option for him.
With Daniel Agger and Jamie Carragher the only defenders of real quality in front of squad players Martin Skrtel and Sotirios Kyrgiakos, Dalglish is looking to add elite depth to his back four. Alex could be the perfect temporary option, as he would make a perfect bridge for a couple of seasons until super prospect Martin Kelly is ready for full-time duty.
Chelsea are expected to ask for around £12 million.