The Swedish delegation headed by foreign minister Carl Bildt's right hand Fredrik Belfrage met recently with Zenawi for bilateral talks, according to a report in the Dagens Nyheter (DN) daily, which cites Ethiopian state radio as its source.
The Swedish foreign ministry has however declined to comment on the reports.
The pair were jailed for 11 years by an Ethiopian court in December after being found guilty of supporting terrorism and entering the country illegally.
The journalists both testified they were in Ethiopia to report on the activities of the Swedish oil company Lundin Petroleum in the Ogaden.
Prior to becoming foreign minister after the 2006 election, Carl Bildt sat on the board of Lundin Petroleum and he has been subjected to criticism for perceived inaction since the journalists' arrest.
Social Democrat Marita Ulvskog has criticised Carl Bildt for not taking the opportunity to meet Zenawi at the recent World Economic Forum in Davos, which they both attended.
Carl Bildt's press secretary has defended the foreign ministry's "quiet diplomacy" approach to the case.
"The discussions which we conduct, when we conduct them and what we say is nothing we publicise as we are convinced that it wouldn't benefit Schibbye and Persson," Anna-Charlotta Johansson told DN.
Schibbye and Persson announced in January that they do not intend to appeal their sentences and instead invest their hopes on the country's "tradition of pardon".
For a pardoning process to commence, the two Swedes will have to pen a letter to the Ethiopian prime minister Meles Zenawi.
They will also need to confess in writing and apologize for entering the country illegally and being in contact with the rebel group ONLF.