However, Brown believes he has star potential and is pursuing a bigger offer, sources told Haynes.
The Celtics have rarely issued extensions to players in the midst of their rookie deals. Rajon Rondo was the last to receive one, agreeing to a five-year, $55-million contract extension in 2009.
If both sides are unable to come to an agreement before Monday's deadline for rookie-scale extensions, Brown will become a restricted free agent following the 2019-20 campaign.
The former third overall pick is set to make $6.5 million this season.
Brown played a major role in Boston's run to the Eastern Conference finals in 2017-18, finishing second in team scoring during the playoffs with 18 points per contest on 46.6% shooting from the floor.
However, he was relegated to the bench after a slow start to the 2018-19 campaign and saw his scoring and 3-point shooting numbers drop from the previous year. Brown eventually regained his starting role prior to the postseason.