Toffees boss David Moyes made his move for the 26-year-old on Sunday after Diniyar Bilyaletdinov completed his transfer to Spartak Moscow for an undisclosed fee, but one which has been reported as £5million.
It is believed a figure has been put to the Govan club which is well below their valuation, believed to be £8million-plus.
With the SPL champions strapped for cash it was thought highly likely that their prize asset would leave in this transfer window, which closes on Tuesday night.
However, Rangers have already rejected a £7million package from West Ham for the former Rapid Vienna player, who has also been linked with Liverpool and QPR.
The Ibrox club now believe there is an even-money chance of Jelavic being a Gers player on Wednesday morning, which will delight manager Ally McCoist who does not want to lose his top goalscorer.
When asked directly about Jelavic today, Moyes said: "I never talk about players at other clubs, I wouldn't do that.
"I don't like it and I didn't like it when I was a player and I kept hearing there were other centre-halves coming in to take my position. I wouldn't talk about players at other clubs.
"If there was any news I would give it to you. You will need to wait until midnight tomorrow night and then we will let you know what is happening."
