Mr Engels was apparently allowed to claim residence in Brussels while living in Berlin in 2012.
There was no immediate comment when the BBC approached Mr Schulz`s SPD party.
However, the party was quoted by Germany`s Spiegel magazine (in English) and Stern magazine (in German) as saying travel arrangements like those for Mr Engels were "normal practice" in Brussels.
The independent EU anti-fraud office Olaf - the same body which investigated French far-right leader Marine Le Pen`s expenses - has told the BBC it is looking at media reports about the conditions of employment of "certain... European Parliament staff members", without naming them.
It said it would analyse these reports to see if it was within its competence to act upon them and to judge whether there was "sufficient suspicion of fraud, corruption or any illegal activity affecting the EU`s financial interests" for Olaf to open a case.
Only when they had been analysed would Olaf decide whether or not to open an investigation, it added.
Mr Schulz chaired the EP from 2012 until this year, when he stepped down to return to Germany for the general election in September.
Under his new leadership, the centre-left SPD, junior partner in Chancellor Angela Merkel`s governing coalition, have seen their poll ratings rise.
Mr Engels now manages his election campaign.
According to a letter (in French) reprinted by Germany`s Spiegel magazine, in January 2012 Mr Schulz asked the European Parliament`s then Director-General for Communication, Juana Lahousse-Juarez, to authorise a "long-term mission" for Mr Engels to support his activities as EP president in Berlin.
In unpublished documents from the EP`s Budgetary Control Committee seen by the BBC, the committee asks EP Secretary General Klaus Welle:
"Why was Brussels chosen as place of employment, which entitled him to 273 daily allowances plus 16% expat allowance plus travel costs, despite the fact that no travel costs occurred? Who decided on the place of employment?"
It also asks by what process Mr Engels was selected for the post.
A formal response to the committee`s questions is expected early next month.
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