Despite both strikers being out of favour at Anfield at present, an ESPN report says Reds boss Jurgen Klopp does not want to lose both this month and leave himself short of cover in the attacking positions.
Southampton and West Ham have been linked with moves for Sturridge, while Newcastle and West Brom are among the clubs who are understood to be keen on signing Ings on loan.
The 25-year-old Ings recently returned to fitness after a serious knee injury and came on as a late substitute in Liverpool's 1-1 draw with Everton on 10 December. He has featured as an unused substitute in the Reds' last two matches.
Sturridge, meanwhile, has only started five Premier League games for the Reds this season, despite being fit for the majority of the campaign.
Reports last month suggested that Sturridge was considering moving to a foreign club on loan in order to boost his chances of making England's World Cup squad, with Spanish clubs Real Betis and Valencia among those monitoring his situation at Anfield.
The 28-year-old has 18 months left on his Liverpool contract, which is potentially worth as much as £150,000 ($200,000) per week.
He has only sporadically featured for Liverpool this season, with manager Jurgen Klopp mostly sticking with his preferred attacking trident of Mohamed Salah, Sadio Mane and Roberto Firmino.
The England striker suffered a fresh muscular injury in late December, but Klopp played down its significance.
"Daniel Sturridge was ill and now [has] a little muscle thing. It's not a big deal, to be honest, but in this period all these things keep you away from games," the German told Liverpool's official website.
Liverpool's next league assignment sees them hosting leaders Manchester City at Anfield on 14 January.