Roma announced the transfers on Friday afternoon.
Defender Mancini arrived from Atalanta on a €2m loan, with the obligation price set at another €13m plus another €8m in bonuses, totalling €23m.
Midfielder Veretout meanwhile arrived from Fiorentina, and was signed for €1m on loan, with the redemption price €16m, plus a further €2m in bonuses.
The pair have been ever presents for Roma coach Paulo Fonseca this season.
— Somos de la Roma (@SomosdelaRoma) February 28, 2020
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