According to Bangkok Post, Mr Wissanu said a complaint may also target Mr Pita’s approval of Move Forward Party memberships.
“In that case, there would have to be a new nationwide general election,” he was quoted as saying.
Bangkok Post also quoted Mr Wissanu as saying that the Election Commission has not taken any action and advised against jumping into “a negative scenario”.
On May 9, Mr Pita said on Twitter he was not concerned about the case because the shares belong to his family’s inheritance fund.
“My status is the manager of the fund and I have consulted as well as informed the National Anti-Corruption Commission a long time ago,” he wrote.
His Move Forward Party emerged as a winner in the May 14 general election, beating big parties such as Pheu Thai with 151 seats in parliament.
Following its historic win, the party formed a coalition with seven political allies, including Pheu Thai, Prachachat, Thai Liberal Party, Thai Sang Thai, Fair, Plung Sungkom Mai and Pue Thai Rumphlang.
Together, they have 312 seats in the 500-member House of Representatives.
For Mr Pita to become prime minister, his party needs the approval from more than half of the 750-seat National Assembly, or at least 376 votes in either the House of Representatives alone or the Senate too.
Several senators remain undecided on whether they will support his premiership, while others have made it clear they will vote against him.