According to reports in Portugal carried by The Metro, the 22-year-old will be offered an extra year extension on top of his current deal as Sporting look to tie him down.
Swap deal offers
The £24m-rated midfielder was the subject of a £13m-plus Joel Campbell offer from the Gunners as the deadline approached, but that was rebuffed by the Portuguese giants.
Manchester United also enquired about a swap deal, possibly involving Nani, but the Portugal international signed for Sporting anyway in a separate agreement as United brought in Angel Di Maria.
Both Louis van Gaal and Arsene Wenger want to resume their pursuits of Carvalho when the January transfer window reopens, but Sporting hope to nip that in the bud before the Premier League clubs get another chance.
If the central midfielder agrees to the terms of Sporting's offer, then the report claims that United or Arsenal would have to pay well above Sporting's previous asking price of £24m.
Wayne Rooney on England
Central midfield priority
Carvalho, who travelled to the World Cup as part of Paulo Bento's Portugal squad, has six caps for his country after a fantastic individual season with his club the previous year.
United explored several options as deadline day approached, with Carvalho their alternative after they came up empty-handed following a protracted Arturo Vidal chase.
Arsenal too were looking to add strength in depth in central midfield, although the £24m valuation put off Wenger, who had already shelled out £32m for Alexis Sanchez.
In any case, another forward was the bigger priority after Olivier Giroud was ruled out until December at the earliest, and Arsenal moved to sign Danny Welbeck from Manchester United for £16m in the end.