Chelsea manager Antonio Conte is keen to bolster his squad with just 11 days of the transfer window remaining.
He is prepared to sell or loan out Diego Costa and wants a striker to come in and compete for places alongside Alvaro Morata and Michy Batshuayi.
The boss is also hoping to strengthen his defence, with Rose being targeted after his public outburst against Tottenham earlier this month.
And the Daily Mail claim the two deals are on Conte's to-do list.
Llorente would likely act as cover if he joined Chelsea, having impressed at Swansea since joining the south Wales club last summer.
He scored 15 goals in 35 games for the Swans last season, including four in his last five games of the campaign.
As for Rose, his future remains uncertain after he criticised Spurs' wage structure, claiming he is keen to play back up north.
However, Rose has not gone as far as issue a transfer request and Spurs believe they will retain the left-back.
Diego Llorente scores superb header against Sevilla
The Daily Mail claim Conte is worried if he doesn't sign his top targets then Chelsea will fail in their mission to defend their Premier League title.
Express Sport understands the Blues are also in for Alex Oxlade-Chamberlain, Alex Sandro, Southampton duo Cedris Soares and Virgil van Dijk, plus Leicester's Danny Drinkwater.