Jennifer Lopez and Marc Anthony inundated with baby gifts

Jennifer Lopez and Marc Anthony have been inundated with baby gifts since arriving home with their newborn twins, a source has revealed.

The new parents reportedly took the babies back to their Long Island home Monday --  and have regularly been receiving multiple deliveries of gifts.

"The twins already got a ton of embroidered pillows with their names on them," says the source -- Anthony and Lopez, however, have not released the names of their son and daughter, who were born Feb 22 -- it has been widely reported, though that their names are Max and Emme.

A spokesperson for Baby K'tan in Fort Lauderdale confirms to American magazine People that pink and blue embroidery were used respectively on the 100 percent cotton carriers and that they "were purchased from us as gifts for both J.Lo and Marc."

In addition, members of the Latino singer's family were spotted shopping at Baby Gap in Americana Manhasset mall last week, confirms a sales rep from the store.