Alexander sustained a broken cheekbone in the last week's victory over former club Rangers.
And Gallacher twisted his ankle in training, making him a doubt for Sunday's Edinburgh derby with Hibernian at Tynecastle.
Hollis, 28, spent four seasons at Well after a six-year spell with Airdrie.
He became a free agent after leaving Motherwell this summer.
"I said earlier in the week that we would be looking at the situation after Neil's injury," Hearts head coach Robbie Neilson told his club website.
"Lee's an experienced goalkeeper and has spent the last few seasons with Motherwell and has played in Europe for them."

Mohammed Sufiyan Choudry and Zaiur Rahman, both members of the Luton chapter of the banned group Al-Muhajiroun, were caught in an undercover police operation in 2015.
Choudry, 23, from Maidenhead, and Rahman, 39, of Luton, were found guilty at the Old Bailey after a retrial.
They will be sentenced at a later date.
The pair were among a group of five covertly recorded when an undercover officer infiltrated their Luton-based group over a 20-month period before police arrested them.
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Bedfordshire Police said the five men attended meetings at a church in Luton and a marquee in Rahman's back garden, during which members praised IS and urged others to support the terror group and travel to Syria to fight.
The meetings were attended by up to 80 people including young children.
The Old Bailey heard that, during a July 2015 gathering, Choudry spoke of "40 trucks driving down Oxford Street full of explosives".
He also said: "A wave is coming. Either be a part of it or you drown.".
The five men were arrested by officers from Bedfordshire Police and the Metropolitan Police Counter Terrorism Command in December 2015.
Three were convicted by a jury in August.
Mohammed Istiak Alamgir, 37, Yousaf Bashir, 36 and Rajib Khan, 37, all from Luton, were convicted for their roles in spreading IS ideology.
Choudry and Rahman faced a retrial after the jury was unable to reach a verdict.
Choudry was found guilty of one count of encouraging support for a proscribed organisation while Rahman was convicted of three counts of arranging/assisting in, arranging/managing a meeting to support a proscribed organisation.
All five are expected to be sentenced together at a later date.

Cwm Taf Health Board and Macmillan Cancer Support are seeking planning permission at the Llantrisant site.
The charity will pay £5m towards the unit.
With eight beds, it would provide inpatient, outpatient and day palliative care for incurable cancer and other conditions.
Macmillan's proposed contribution, which is subject to planning approval, would be its biggest single investment in Wales.
Dr Ian Back, consultant in palliative medicine at Cwm Taf University Health Board, said: "A new state-of-the-art specialist unit would provide patients with the high-quality care they need in a comfortable environment for both them and their loved ones.
"Building the new unit at Royal Glamorgan Hospital will also ensure that patients have access to specialist tests and treatment more easily without the need for unnecessary transfers between hospitals when they are often too unwell."
Palliative care for cancer patients in the county is currently provided at Pontypridd Cottage Hospital.
This means patients need to be transferred to Royal Glamorgan Hospital to access specialist tests and treatments, which the health board said can be "difficult and distressing".
Susan Morris, head of services for Macmillan Cancer Support in Wales, said: "After extensive consultation with staff and patients, the unit would be designed using Macmillan's expertise and understanding of the needs of people with cancer and their families, including a kitchen and communal areas where they can spend time together."
If the plans are approved, work is expected to start later this year with the centre potentially opening in late 2018 or early 2019.
Hearts have signed former Motherwell goalkeeper Lee Hollis on a short-term deal following injuries to Neil Alexander and Scott Gallagher.
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Two men have been convicted of organising terrorist speeches and encouraging support for so-called Islamic State (IS).
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Plans for a £6.75m specialist cancer care unit at Royal Glamorgan Hospital in Rhondda Cynon Taff have been unveiled.