The west London side is reportedly prepared to pay £4 million to release Lampard from his contract with Championship side Derby County, where he spent his first year as a manager.
A three-year contract is waiting for Lampard at Stamford Bridge, according to Twomey.
Chelsea owner Roman Abramovich identified the 41-year-old as the ideal replacement for Maurizio Sarri, who departed for Juventus after a single season in England.
Lampard would find several familiar faces at Chelsea, including old teammate Petr Cech, who returned to the club as a technical and performance adviser on Friday.
The former midfielder became the club's all-time leading scorer during his 13 years with the Blues.