Jürgen Klopp offered what would be a world record fee for the 25-year-old although it falls short of the Italian club's asking price by some £4.6 million.
Chelsea are aware that Thibaut Courtois could leave this summer after holding off extending his contract at Stamford Bridge.
Although Chelsea are expected to make a rival offer to Roma for Alisson, The Times report that the 25-year-old has already indicated that his preference is to join Liverpool.
Liverpool make €70 million Alisson swoop - reports - Football
Alisson's valuation by Roma is more likely to be met by Chelsea.
Klopp found himself forced to defend the German after he made another blunder as Liverpool edged out Tranmere in last Tuesday's friendly.
This particular transaction, if it happens, would be record setting as Roma shot stopper Alisson Becker could become the world's most expensive goalkeeper. The 25-year-old initially made an encouraging return to the side but his two terrible errors in the Champions League final undermined his first choice claims and he made another mistake in last week's pre-season friendly at Tranmere Rovers. The Liverpool manager has also considered Danny Ward during pre-season but is edging closer to addressing a long-standing issue at Anfield with the bid for Alisson.
Trump hits the links at his Scottish golf resort
Trump's history with Scotland traces to his mother, who grew up on the Scottish Isle of Lewis in a village called Tong. A Greenpeace demonstrator paraglides flies over a Trump-owned golf course in Scotland during the president's visit.