The Brazil international has moved to the Nou Camp for a fee which could rise to £142m, leaving the Reds with money to spent in the remaining weeks of the January transfer window.
Monaco's Thomas Lemar and Riyad Mahrez of Leicester City have been heavily linked with moves to Merseyside but Aldridge wants to see Sanchez arrive at Anfield.
"They need to spend the money now and make sure it is reinvested into the squad," he told the Irish Independent. "And they should do whatever it takes to bring in Alexis Sanchez. In a dream world, I would have done a straight swap for Luis Suarez but that won't happen.
"But Sanchez is a player I would like to see at Anfield. He has a lot of baggage around him, but he is a top-class player. Liverpool wanted him when he was at Barcelona, he came to Arsenal and has done really well.
"Knowing that Sanchez's contract is up in the summer, I would have offered Arsenal £40m, plus one pound, as they did with Luis Suarez. I know there is a history with Sanchez and he has that baggage but you can't doubt his class.
"The one thing that could really prevent a deal like that for Liverpool is if Manchester City come in for Sanchez. Liverpool have cash to hand right now but with City, they deal in Monopoly money and it's difficult to get a player if City really want him."
The Chile international came close to joining Liverpool as part of the deal which took Luis Suarez to Barcelona, but he eventually opted to move to north London.
Sanchez has enjoyed great success with Arsenal but has entered what will be the final six months of his tenure with a move to Manchester City expected to happen either this month or in the summer.
If as anticipated Sanchez does move to the Etihad Stadium, then Lemar and Mahrez remain the outstanding candidates to step into the void left by Coutinho.
And Aldridge would welcome either player this month but would prefer to see Algeria international Mahrez arrive from Leicester, given his Premier League experience and eligibility for the Champions League.
"I have heard talk about Thomas Lemar, at Monaco, I haven't seen too much of him but the reports are very good," he added. "He is the right age, in terms of what the American owners at Liverpool like. They like to buy young players, Lemar is 22 and that's ideal for Liverpool right now, though a lot of teams are after him and Monaco have said they want to keep him.
"There's also talk about Riyad Mahrez, who plays a very similar role to Coutinho. Mahrez had a number of things going for him: he has played in the Premier League before and he is eligible for the Champions League, and those factors could be important, he could be a good replacement. Should they even look to sign two players now as the money is there? Champions League is key and having a new player who is eligible, as Mahrez would be, could prove vital."