The 30-year-old is surplus to requirements at Villa Park and spent the second half of last season on loan at Reading.
QPR are also keen on Elphick, who would bring some much needed experience to a Tigers backline that has already conceded 14 goals in seven league and Cup outings this term.
On the hunt for new faces, Adkins said today: “I have highlighted that we have got to strengthen and we have got until 5pm tomorrow.
“We are working hard behind the scenes. There is money on the table for players for loan deals that then have the potential to turn into permanent deals.”
City, meanwhile, could be without Jackson Irvine and Nouha Dicko against Derby County on Saturday due to injury.
Adkins said: “It’s looking like Nouha and Jacko will be struggling for the weekend. Nouha is feeling his knee a little bit so he dropped out of training today.
“He played the full 90 minutes on Tuesday so we will just see. If the game was now he wouldn’t be fit but we have still got Friday and Saturday to recover.
“Hopefully, it is nothing too serious but we’ll wait and see. Jackson has not been able to join in so he has to be classed as a big doubt for the weekend. I’m not ruling him out but he’s a big doubt.”