The 23-year-old striker is the Reds’ first signing of the month, with Joe Cole’s return to West Ham set to be confirmed in the coming days.
Liverpool have been linked with a host of players in recent months, including the likes of Atlético Madrid’s Arda Turan and Blackpool winger Thomas Ince.
But although Rodgers accepts that he will need to strengthen his squad further down the line, the Reds boss has played down the chances of their being many more arrivals on Merseyside this month.
“There is nothing in the pipeline,” said Rodgers. “It was never going to be a busy month for us and there are not too many players available who could come in and help and improve what we’re doing.
“We’re not going to spend money needlessly but there is no doubt over the coming months we need to strengthen the squad. That will come with time.”
Rodgers’ Liverpool side moved up to eighth place in the Premier League table following Wednesday’s 3-0 win over Sunderland at Anfield.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news