Clarets announce retained list
But Michael Kightly and Joey Barton have both been released.
Boss Sean Dyche has confirmed on-going conversations are taking place with the representatives of Boyd, 31, and former England international Robinson, 37.
Boyd made 36 appearances in the Premier League this season as he helped the Clarets sustain their top-flight status, while Robinson played three times as back-up to club captain Tom Heaton.
Kightly – who recently ended a successful loan spell at Burton Albion – is to be released after three seasons with Burnley.
The 31-year-old previously spent a season on loan from Stoke City, helping the Clarets win promotion from the Championship in 2013/14, before completing a Â£1.5m switch.
Kightly also contributed to last season’s Championship title-winning success and made a total of 88 appearances for the Clarets.
The former Wolves wideman scored four goals in 12 Championship appearances for Burton after signing for the Brewers in January, and played a key role in helping Nigel Clough’s side stay in the division.
Barton – who joined the Clarets for a second spell in January and made 14 Premier League appearances up to the end of April – is being released following his recent suspension by the Football Association for historic betting offences.
Dyche said: “Joey leaves us for the obvious reasons after he has again been a big part of what we have achieved this season.
“Equally, Kights has been a great servant of the club and always a tremendous person to have around the group.
“Both players have had a big effect on what we have done during their time at Turf Moor and we wish them well, as we do all those leaving this summer.”
Defender Jon Flanagan has also left the club after returning to Liverpool following his season-long loan.
The Clarets’ retained list shows four members of the Development Squad are being released at the end of their contracts.
Forwards George Green and R J Pingling, midfielder Christian Hill and defender Taofiq Olmowewe will all leave Turf Moor.
But the club has taken up a further year’s option on the contracts of defender Luke Hendrie, 22, who has been a regular on loan with Scottish Premier League side Kilmarnock this season, defender Alex Whitmore, 21, who spent a season-long loan with Morecambe, full-back Arlen Birch and goalkeeper Conor Mitchell.
Striker Rouwen Hennings is expected to return to Burnley following a loan spell with German second-tier side Fortuna Dusseldorf as he enters the final year of his contract at Turf Moor.