The Toffees have been busy in the market this summer, bringing in Steve Naismith, Steven Pienaar and Kevin Mirallas.
FC Copenhagen full-back Bryan Oviedo is set to sign for the club on Friday after having a work permit approved, and Round has confirmed that the Merseyside club will look for extra reinforcements following the departure of Jack Rodwell to Manchester City.
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"We are only ever going to get one or two players at this late stage, and if we can bring in just one today I think that will be abonus.
Round cited Oviedo's potential versatility as attractive to manager David Moyes and insisted that the Scottish manager was continuing to work hard to add another new signing.
"Bryan is the closest we have got," Round added. "And other than that, the manager is going to be working all day I think with the chairman [Bill Kenwright] to try to bring somebody else in."
"He [Oviedo] can play left back, left midfield, left wing - all the way down the left hand-side. That will give us good cover in our squad."