Mauro Contini, the mayor of Quartu Sant'Elena, the Cagliari district where the stadium is located, was also arrested along with another local official by the Forestry corps, an agency of the Italian police, the source said.
They were charged with embezzlement and false representation in the rebuilding of the stadium.
Gazzetta dello Sport showed a photograph of Cellino standing in front of a blue gate which it said was the Buoncammino prison in Cagliari.
The Sardinian club issued a statement of support for its 56-year-old president who has been at the helm since 1992, apart from a break in 2005-06.
"Officers, employees, technical staff and players are close to their president Massimo Cellino in these difficult times," said the statement.
"We have faith in the work of the judiciary and are confident that the investigation will show the non-involvement of president in the alleged facts.
"At this time we would like to say just one thing: 'Strength to the president, we are with you'."
Cagliari moved to the small Is Arenas stadium, which had previously hosted third-tier matches in the 1980s, at the start of this season but, with only one permanent stand, they have had trouble from the outset.
Three temporary stands were built on the other sides of the pitch but their opening match against Atalanta was staged behind closed doors after local authorities ruled the stadium was not ready.
Their following match against Roma was cancelled by the Cagliari city government after the club defied its orders to play without supporters.
On that occasion, Cellino told fans the ground was safe, invited them to turn up for the match anyway and slammed the city government for placing bureaucratic obstacles in the club's way.
Roma were awarded a 3-0 win but Cellino won a court case on Thursday which could lead to the game having to be played after all.
Less high profile matches proceeded at the Is Arenas but Cagliari were banned from hosting a game against champions Juventus in the stadium and had to switch that fixture to Parma.
Last Sunday's game against AC Milan was initially set to be moved to Turin amid security worries but eventually went ahead at the stadium after a regional tribunal ruled it was safe.
Cagliari had played at the Stadio Sant'Elia from 1970 until around halfway through last season but abandoned the arena due to disagreements with the local authorities and over growing safety worries.
They played their remaining matches last term in Trieste, near the border with Slovenia and nearly 1,000km away.