Palace have been desperate to add a young right-back to their squad since failing to replace Aaron Wan-Bissaka following his £50million move to Manchester United last summer.
The Eagles identified Walker-Peters as a target during the summer window but could not come to an agreement with Spurs over a sale then and have been discussing a loan move this month.
Southampton - who will lose Portuguese full-back Cedric at the end of his contract this year - are also interested.
However, Hodgson has suggested that Spurs have not yet decided whether they are happy to let the 22-year-old leave in January.
"[There is] no update," said Hodgson. "There was initial contact I think between the clubs and the suggestion was made and one I was behind.
"It would have been a good move for us I think, but as far as I know nothing is happening and I'm not sure what Tottenham’s intention with Kyle Walker-Peters is."
While things are moving slowly between the two clubs, Hodgson remains keen to bolster his squad with up to four more signings this month.
Cenk Tosun has arrived on loan from Everton until the end of the season and Palace remain interested in adding another striker to their ranks.
Cover at right-back is also required, as is cover for Patrick van Aanholt down the left.
"Going into the window we were hoping for two strikers and full-back cover at right and left-back," said Hodgson.
"Potentially extra cover in midfield, maybe a different type of player to the player Victor [Camarasa, whose season-long loan was cut short] was and someone who could take his place in the squad.
"We were looking to research those possibilities, but nothing has happened so far other than the fact we have got Cenk Tosun on loan from Everton."