The England pair are being tipped to leave the Emirates Stadium, with Southampton said to be keen on Oxlade-Chamberlain in particular.
And the report lists Burke as a potential replacement, having been under observation from Arsenal's chief scout Steve Rowley.
Burke made his Forest debut in September 2014 but has impressed in a breakthrough campaign at the City Ground, signing a new four-and-a-half-year deal in February.
The 19-year-old has scored twice in 16 Championship outings for the Reds, and has been a rare shining light in an otherwise disappointing season.
Burke's form attracted the attention of Scotland boss Gordan Strachan last month, with the 6ft 2in winger going on to make his international debut in the friendly against Denmark.
Since exiting the top flight 17 years ago, Forest's academy has spawned a number of Premier League stars including, notably, Leicester captain Wes Morgan and former England internationals Jermaine Jenas and Michael Dawson.