The ad that portrays Abbas and Meshaal locked in an embrace, will be posted in 10 European papers in the coming weeks, Xinhua reported citing the Ynet News website.
Abbas is expected soon to meet the Hamas chief in Cairo to resume reconciliation talks.
However, Lieberman's decision has been met with opposition from the ministry and the embassies of the countries where the ad is expected to be published.
"This is an initiative of the minister and we don't deal with it," a spokeswoman at the foreign ministry told Xinhua, dismissing its involvement in the campaign.
Lieberman's spokesman also refused to comment on the campaign.
Reports also said Lieberman has decided to ask Jewish organisations to post the ads using their names instead of the foreign ministry, in a bid to avoid controversy.
Both Lieberman and Israeli Prime Minister Benjamin Netanyahu have criticised Abbas' reconciliation attempts with Hamas, saying it would harm the peace process.
"To my regret, he (Abbas) strives for unity with the same Hamas that is supported by Iran," Netanyahu said.
"It's interesting that Abbas has issued no condemnation of the remarks about the destruction of Israel," Xinhua quoted Netanyahu as saying.