The departure of former B&Q chief executive Toogood is was announced to staff earlier this week, according to the Financial Times.
The paper added that Icelandic bank and Ilva's debt provider Kaupthing was in discussions with Advent to buy its stake in the business.
Advent paid an estimated£60 million for Ilva in 2003 and has since seen the company embark on an aggressive expansion into the UK.
It has large sites in Gateshead, Manchester and Thurrock and last year said it planned to open as many as 20 UK stores in 10 years.
However, the paper said that Ilva's UK businesses had generated poor results and that this had prompted Advent to look for an exit from the company.
It is unclear how this upheaval will affect the planned UK expansion.