Lampard, the Blues’ record goalscorer, has been strongly linked with the job since Maurizio Sarri left the role this month.
A statement issued by Derby said: “Derby County Football Club can confirm that they have granted permission for Chelsea Football Club to speak to Frank Lampard about the vacant managerial position at Stamford Bridge.
“With pre-season fast approaching for both clubs it is hoped this will allow Chelsea to swiftly conclude their discussions.
“The club will make no further comment until it is appropriate to do so.”
While missing out on promotion was hugely disappointing, it represented a good first season in management for the 41-year-old – who has since been in talks about a new contract with the Rams.
Chelsea would have to pay compensation to take Lampard away from Pride Park as he has two years remaining on the deal he signed last summer.
The former England midfielder made 648 appearances for the Blues, scoring 211 goals and winning 11 major trophies.
If he does take over at Stamford Bridge, he will inherit a transfer embargo after Chelsea were given a two-window ban by Fifa – a decision the club are appealing against at the Court of Arbitration for Sport.
Frank Lampard managerial timeline
Here, we pick out some of the key events that have punctuated the 41-year-old’s first season in management:
May 31, 2018 – Derby announce the appointment of Lampard as manager, his first managerial role, on a three-year contract.
July 17 – Lampard signs young loan players Mason Mount, from Chelsea, and Liverpool’s Harry Wilson.
August 3 – Lampard’s first match in charge sees Tom Lawrence score a last-minute winner in a 2-1 victory at Reading.
August 11 – A first defeat swiftly follows in Lampard’s second match, a 4-1 loss at home to Leeds.
September 25 – Derby knock Manchester United, and Lampard’s old Chelsea boss Jose Mourinho, out of the Carabao Cup on penalties following a 2-2 draw at Old Trafford.
December 29 – Jack Marriott grabbed a dramatic late winner as the Rams recovered from 3-2 down with five minutes remaining to beat Norwich 4-3.
January 10 – Police are called to Derby’s training ground following reports that a man was “acting suspiciously” outside. Leeds manager Marcelo Bielsa later admits sending someone to spy on their next opponents.
January 11, 2019 – Lampard described Bielsa’s pre-match conduct as “unethical” after Leeds beat the Rams 2-0 at Elland Road to go five points clear at the top of the Championship.
May 5 – A 3-1 win over West Brom secured a sixth-place finish for Derby and a play-off against none other than Leeds.
May 11 – Kemar Roofe’s goal gives Leeds controversial a 1-0 first-leg win at Pride Park. Derby were awarded a penalty which was then overturned on the advice of the assistant referee, a decision Lampard called “incredible”.
May 15 – Lampard gets his revenge, and Derby’s social media team are quick to rub it in, with a thrilling 4-2 win at Elland Road to secure a place in the play-off final.
May 27 – Derby fall at the final hurdle and miss out on promotion to the Premier League after a 2-1 Wembley defeat by Aston Villa. Afterwards Lampard refuses to be drawn on the Chelsea speculation.
June 25 – With Maurizio Sarri having vacated the Chelsea job on June 16 to join Juventus, Derby announced that the Blues had been granted permission to speak to Lampard about the role.