The England and Wales Cricket Board (ECB) declined the request in order to allow the 30-year-old to rest.
Broad bowled 21 overs in Notts' opening match, taking four wickets as they beat Leicestershire by 10 wickets.
He and fellow paceman James Anderson, who is unavailable for Lancashire, will return for the third round of games.
"Both Stuart and Jimmy are committed to their counties and keen to play as much possible," said ECB director of cricket Andrew Strauss.
"But equally they recognise both the challenges and demands to come over the summer. Counties were made aware of the likely schedule back in January."
Despite losing the services of the centrally contracted Broad, Notts will have another England pace bowler, Jake Ball, available after the ECB rested him for the first game.
England do not play a Test until 6 July, but then face seven Tests in just over two months against South Africa and West Indies, before seven more in Australia and New Zealand over the winter.
England bowler Stuart Broad will not be available for Nottinghamshire's County Championship match at Durham on Friday, despite a request from his county.