An "environmentally friendly" building with a cafe, gallery and shop is planned for the park near Wakefield.
Peter Murray, executive of the YSP, said it would provide an "important new element" of the park's infrastructure and increase its income.
Arts Council England has given £1.7m to the park and the remainder is to come from fundraising.
A planning application for a second visitor centre has been submitted to Wakefield Council and it is due for completion in late 2017.
The new building has been designed to make the minimum impact on the site, YSP said.
The latest project on the 500-acre estate follows the development of the main visitor centre in 2002, the underground gallery in 2006 and refurbishment of the estate chapel in 2011.
The park attracted more than 400,000 visitors in 2014 the highest number since it opened in 1977, it said.
The development plans were revealed as the park's latest sculpture Wave is being installed.
It is part of the installation Blood Swept Lands and Seas of Red that was seen at the Tower of London in 2014 where 888,246 poppies honoured every death in the British and Colonial forces of World War One.
Yorkshire Sculpture Park (YSP) has unveiled plans for a new £3.8m visitor centre at its Bretton Park entrance.