The Daily Mail claims Johnson's £18.5 million move from Portsmouth means Rafa Benitez will now have to bring in funds to continue his rebuilding plans.
The paper says Madrid president Florentino Perez has offered £30m for Alonso after Liverpool last week rejected a bid of £20m for the Spain international.
Alvaro Arbeloa could also be the subject of interest from Madrid following Johnson's arrival, with the Spaniard likely to struggle for a regular starting place.
At the weekend, Alonso said the pound's weakness against the euro made a return to Spain more attractive.
'Of course we footballers think about it; we are not stupid, we keep an eye on it," Alonso said. "When you see your contract down by 30 per cent you cannot be happy. I cannot do anything – it must be Gordon Brown or David Cameron.
I think the weakness of the sterling is not helping the Premiership because for those competing and fighting against the European teams it is a big weak point. Hopefully over the next few months and years it will get back to what it was not so long ago."
His Anfield colleague Fernando Torres admitted he was concerned about losing his Spain team-mates Alonso and Arbeloa, but insisted that playing for a club with the traditions of Liverpool was more important than money.
"I worry about losing my team-mates, of course," Torres said. "But when you play for a team like Liverpool you don't have to worry about that. You are a top player and you will get a good contract. You don't have to be scared to lose a little bit of money."