Critics have pointed to her lack of policy experience and her apparently relatively poor English language skills (although fluent in German), which will no doubt be important since all designates (expect Ms Jeleva) declare themselves fluent in English...
It is perhaps with these considerations in mind that her original allocation to the consumer protection dossier was revised to instead nominate her for the entirely new portfolio of International Co-operation, Humanitarian Aid and Crisis Response - which is certainly seen as one of the least important dossiers on offer.
And if that wasn't enough, reports in the European press have brought to light allegations of her husband having connections with organised crime (the respected German paper Die Welt even referred to her recently as the "gangster bride") and her past position as the owner and/or manager of a trading company called Global Consult and another called Auto Spa, which she is alleged to have not declared.
Bulgarian legislation stipulates that the MEPs are not entitled to conduct any other sort of paid activities in addition to their work in Brussels, and the debate into a potential conflict of interest raged upon her election to the Parliament as an MEP in 2007.
Sensing that MEPs will look to delve into these undeclared financial interests and her personal and family background, Commission president José Manuel Barroso has apparently taken the unprecedented step of sending a letter to MEPs telling them that any accusations levelled against the Bulgarian EU commissioner nominee must be backed up by proof.
"Anyone accusing anyone of any wrongdoing should of course present corresponding evidence, as in our democratic societies the rule of law implies that the accuser should prove accusations," he writes in the letter.
"All the commissioner-designates' declaration of interests have been looked at carefully in terms of respect for the terms of the Code of Conduct. Ms Jeleva confirmed to me that her declaration, like those of other commissioners, respects the Code."
Speaking ahead of the hearings in a press conference on Monday morning, Andrew Duff MEP (ALDE, UK) said that "we are all conscious of the press speculation, especially in the German press, on the family's business concerns. And I am certain that the hearing of her is to be focussed upon some of those issues."
Mr Duff, who is charged with overseeing how the hearings are carried out, also said that MEPs would not hesitate to ask Mr Barroso for more information if they felt that a candidate was not forthcoming enough.
With the Parliament only permitted by the Treaties the opportunity to vote on the entire Commission and not on each individual candidate, Mr Barroso will be keen to ensure each candidate is to Parliament's satisfaction.
It will therefore be interesting to see how Ms Jeleva's Hearing goes this afternoon and to what extent members will interrogate her on these matters in light of Mr Barroso's letter.
To watch the proceedings yourself, click here. (Hearing runs from 3.30-6.30pm GMT).