The Bible says that you need to believe to be saved (e.g. Eph. 2:8). Animals are not able to believe, so animals cannot be saved.
However, there will be animals in God’s kingdom. The prophet Isaiah has several description of God’s kingdom that include animals:
6 The wolf shall dwell with the lamb,
and the leopard shall lie down with the young goat,
and the calf and the lion and the fattened calf together;
and a little child shall lead them.
7 The cow and the bear shall graze;
their young shall lie down together;
and the lion shall eat straw like the ox.
8 The nursing child shall play over the hole of the cobra,
and the weaned child shall put his hand on the adder’s den. (Isaiah 11:6-8)
The wolf and the lamb shall graze together;
the lion shall eat straw like the ox,
and dust shall be the serpent’s food.
They shall not hurt or destroy
in all my holy mountain,”
says the Lord. (Isaiah 35:25)