"Hello Ronaldo," the Gazzetta dello Sport headlined a report saying that Allegri had called Ronaldo on Monday, suggesting the transfer deal was all but done, AFP reported.
No details of the conversation were released by the newspaper which said only that it was "friendly" while other reports said he had discussed tactics.
La Gazzetta said that Ronaldo's agent Jorge Mendes was expected in Madrid on Tuesday when Real directors are scheduled to meet.
However, rival sports paper Corriere dello Sport said the directors meeting had been put off so that "nothing would be revealed" about the deal.
"The Cristiano Ronaldo affair has become a game of chess," the newspaper said.
However, details about transfer fees were still unclear with Gazetta citing €120 million ($140.5 million) and the Corriere putting the deal at between €130m and €150m.
Juve is ready to sign the five-time Ballon d'Or winner on a four-year deal for an annual basic salary of €30 million.
Ronaldo, who joined Real in 2009, has complained that his Madrid salary of €23.6 million is too low.