New injury Head-ache for Australia in Sri Lanka
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Smith missed the game with a quad problem and Australia were also without Marcus Stoinis, Kane Richardson, Mitchell Starc and Ashton Agar.
"There is a lot of them," Finch said, adding that he was "not sure" if Head, who scored a useful 70 not out in the third ODI on Sunday, would be fit for the two-match Test series beginning on July 29 in Galle.
"Especially where he fields in the outfield, he does quite a lot of kilometres and the ground is quite heavy," Finch said, adding that Head's absence was a "bit more of a precaution."
He said he doubted Smith would be available either.
"But that could change. I mean he'll be doing training this afternoon. We will go down there and have a look at the conditions and everything. But from what I understand, he won't be available for tomorrow," Finch said.
Finch also said that Starc was "still not right" with his finger, which he cut on the spike of his boot in the first ODI.
"Where it is, the stitching and things like that (are) just not quite 100-percent healed. Without being able to bowl with tape on, he will be out for tomorrow as well," Finch said.
Finch blamed the "disappointing" series defeat on Australia's inability to build partnerships but said the games were still good preparation for the Tests.
"Any time we've started to get any momentum in the game, we lose a wicket and in these conditions, once you lose one, they go back-to-back pretty quick," Finch said.
He also defended international bilateral series in the age of domestic T20 tournaments like the Indian Premier League, which have registered huge growth in funding and popularity in the last few years.
Finch said that the standard "isn't the same when you play domestic competitions" and international bilateral series were still "important to the game".
"Most satisfying for me was the progress of a team as we've made through this journey so far, just to see obviously the tactics coming together, to see guys out in the middle scoring runs, to see how we've chased totals down," Silverwood told reporters.
"And to me, one of the most exciting things is but we're still nowhere near the potential that we have in that team... If the guys keep learning and keep progressing the way they are, then we will be... in a very strong position to challenge anywhere in the world."