Allan joined Pompey on an initial month-loan loan in February, but Appleton is keen to keep the 20-year-old at the club for the rest of the season.
“He’s a player we were following when I was assistant manager at West Brom,” Appleton said in the Birmingham Mail.
“It’s a shame he’s a West Brom player, long term. But I think in the short term, hopefully, we can keep hold of him until the end of the season and he can have some effect on what happens to Portsmouth Football Club from now until then.
“Because of the situation we’re in, we will just have to continue the loan month by month.”
Allan joined West Brom from Scottish side Dundee United in the January transfer window.
