Liverpool are this week set to step up their efforts to appoint Hodgson to succeed Rafa Benitez.
But Hodgson is yet to give any assurances he will go to Anfield and has told friends that he would prioritise England over any club job.
England must beat Slovenia on Wednesday to be sure of progressing through the group stages of the World Cup.
If they fail, it is expected boss Fabio Capello would either quit or be sacked.
Former Inter Milan boss Hodgson would be the FA's first-choice replacement for Capello and is said to be watching developments carefully, but his hand could be forced if Liverpool demand a decision this week